# Low-dimensional magnetism of BaCuTe2O6

P. Bag  N. Ahmed  Vikram Singh  School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India  M. Sahoo  Department
of Physics, University of Kerala, Kariavattom, Thiruvananthapuram-695581,
India  A. A. Tsirlin<EMAIL_ADDRESS>Experimental Physics VI, Center for
Electronic Correlations and Magnetism, Institute of Physics, University of
Augsburg, 86135 Augsburg, Germany  R. Nath<EMAIL_ADDRESS>School of
Physics, Indian Institute of Science Education and Research
Thiruvananthapuram-695551, India

###### Abstract

X-ray diffraction, thermodynamic measurements, and density-functional band-
structure calculations are used to study the magnetic behavior of BaCuTe2O6, a
member of the $A$CuTe2O6 structural family that hosts complex three-
dimensional frustrated spin networks with possible spin-liquid physics.
Temperature-dependent magnetic susceptibility and heat capacity of the Ba
compound are well described by the one-dimensional spin-$\frac{1}{2}$
Heisenberg chain model reminiscent of the Sr analog SrCuTe2O6. While the
intrachain coupling $J/k_{\rm B}\simeq 37$ K is reduced compared to 49 K in
the Sr compound, the Néel temperature increases from 5.5 K (Sr) to 6.1 K (Ba).
Unlike the Sr compound, BaCuTe2O6 undergoes only one magnetic transition as a
function of temperature and shows signatures of weak spin canting. We
elucidate the microscopic difference between the Sr and Ba compounds and
suggest that one of the interchain couplings changes sign as a result of
negative pressure caused by the Sr/Ba substitution. The Néel temperature of
BaCuTe2O6 is remarkably insensitive to the magnetic dilution with Zn2+ up to
the highest reachable level of about 20 %.

##  I Introduction

Frustrated magnetic networks host unusual ground states and non-trivial
dynamical phenomena that become intricate especially in the case of quantum
spins [1, 2, 3]. Compounds based on the spin-$\frac{1}{2}$ Cu2+ ions offer one
of the best playgrounds for the experimental realization of such networks.
Different types of crystal structures give access to multiple interaction
geometries, including kagome lattices of interacting spins [4, 5]. Layered
kagome compounds can even be found in nature as copper minerals [6], but their
three-dimensional analogs – material realizations of the hyperkagome spin
lattice – remain scarce. The only example in the cuprate family is PbCuTe2O6
[7, 8], where signatures of a three-dimensional quantum spin liquid with
fractional spinon excitations were recently reported [9].

PbCuTe2O6 has a fairly complex cubic crystal structure. The hyperkagome
lattice formed by second-neighbor interactions is augmented by interactions
between first and third neighbors [9]. When Sr substitutes for Pb, the crystal
structure and even the cubic lattice parameter remain nearly unchanged, but
the magnetic behavior is modified drastically. SrCuTe2O6 is a spin-chain
compound with a robust magnetic order at low temperatures [10, 11, 12].
Predominant third-neighbor interactions give rise to spin chains along each of
the three main cubic directions [10]. The presence of multiple phase
transitions [10, 13], complex temperature-field phase diagram [12], and non-
collinear magnetic order [11, 12] witness frustrated interchain interactions
that may be residues of the strong frustration present in PbCuTe2O6.

The drastically different behavior of the Pb and Sr compounds suggests an
extreme sensitivity of this structure type to chemical substitutions. This
fact motivated us to prepare and explore BaCuTe2O6, another member of the same
family, and attempt its further tuning by Zn doping that may introduce both
chemical pressure and magnetic dilution.

BaCuTe2O6 retains the cubic crystal symmetry and overall structural geometry
of the $A$CuTe2O6 compounds, but shows a notable lattice expansion, with the
room-temperature cubic lattice parameter increasing from $a=12.473$ Å [14]
(Sr) and $a=12.49$ Å [7] (Pb) to $a=12.87$ Å (Ba). We demonstrate that,
despite this effective negative pressure, BaCuTe2O6 strongly resembles
SrCuTe2O6 and hosts weakly coupled spin chains due to predominant magnetic
interactions between third neighbors. Interestingly, substituting Ba for Sr
decreases the intrachain coupling but increases the Néel temperature. We
combine this observation with an ab initio microscopic analysis to identify
relevant magnetic interactions and their changes upon chemical substitutions.
We also show a remarkable insensitivity of the Néel temperature to magnetic
dilution as another experimental witness of the underlying frustration in this
family of compounds.

##  II Methods

Polycrystalline samples of Ba(Cu1-xZnx)Te2O6 ($x$ = 0 – 0.20) were synthesized
following the same procedure adopted for the synthesis of SrCuTe2O6 [10]. The
initial reactants BaCO3 (Aldrich, 99.995%), CuO (Aldrich, 99.999%), TeO2
(Aldrich, 99.995%), H2TeO4.2H2O (Alfa Aesar, 99%), and ZnO (Aldrich, 99.99%)
were taken in stoichiometric ratios required for the synthesis of BaCuTe2O7
and its Zn doped samples. For $x=0$, the reactants were ground thoroughly,
pelletized, and fired at 670 $\degree$C for five days in flowing argon
atmosphere with several intermediate grindings. For the doped samples, the
firing temperature was varied from 670 $\degree$C to 720 $\degree$C.

Similar to SrCuTe2O6, the flowing argon provides a reducing atmosphere which
leads to the loss of oxygen and hence the formation of BaCuTe2O6 instead of
BaCuTe2O7. The powder x-ray diffraction (XRD) pattern was recorded at room
temperature using a PANalytical powder diffractometer (CuKα radiation,
$\lambda_{\rm avg}\simeq 1.5418$ Å). The samples were found to be single-phase
up to $x=0.18$, while for $x=0.20$ BaCuTe2O7 emerged as a minor impurity
phase. The amount of the impurity phase for $x=0.20$ was found to be very
small (about $\sim 2$%). Our repeated attempts to achieve higher doping levels
by increasing or lowering the firing temperature were unsuccessful.

Figure 1:  Synchrotron x-ray diffraction data of BaCuTe2O6 at 300 K and 80 K.
The solid lines denote the Rietveld fits of the data. The Bragg peak positions
are indicated by green vertical bars, and the bottom blue line shows the
difference between the experimental and calculated intensities. The refinement
residuals $R_{I}/R_{p}$ are 0.028/0.092 at 300 K and 0.016/0.076 at 80 K.
Figure 2:  Left panel: Crystal structure of BaCuTe2O6, with black curved lines
denoting the spin chains. The blue square planes are the CuO4 plaquettes,
while the TeO3 units are shown in green color. The Ba atoms are omitted for
clarity. Right panel: spin lattice of BaCuTe2O6 composed of crossed spin
chains (solid red lines) of Cu2+ ions. The dominant interchain couplings
$J_{2}$ and $J_{6}$ forming frustrated loops are also shown. VESTA software
was used for crystal structure visualization [15].

For structure determination, high-resolution synchrotron XRD data were
collected at the ID22 beamline of ESRF using the wavelength of 0.4107 Å. The
powder sample placed into a thin-wall borosilicate capillary was cooled down
with liquid-nytrogen cryostream and spun during the measurement. Diffracted
signal was measured by 9 scintillation detectors, each preceded by a Si (111)
analyzer crystal. Rietveld refinements of the XRD data were performed using
JANA2006 and FULLPROF software packages [16, 17].

Magnetic susceptibility $\chi$ ($\equiv M/H$, $M$ is the magnetization and $H$
is the applied field) was measured as a function of temperature (2 K $\leq
T\leq 350$ K) and magnetic field (0 T $\leq H\leq 9$ T) using a vibrating
sample magnetometer (VSM) attachment to the Physical Property Measurement
System (PPMS, Quantum Design). Heat capacity $C_{\rm p}$ was measured on a
small piece of the sintered pellet via the relaxation technique using the heat
capacity option of PPMS.

Exchange parameters were calculated for the spin Hamiltonian

$\mathcal{H}=\sum_{\langle ij\rangle}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j},$ (1)

where the summation is over lattice bonds $\langle ij\rangle$. Exchange
integrals $J_{ij}$ were extracted via density-functional (DFT) band structure
calculations performed using the FPLO code [18] within generalized gradient
approximation (GGA) [19] for the exchange-correlation potential. The mean-
field GGA+$U$ approach was used to treat correlation effects in the Cu $3d$
shell, with the Hubbard repulsion parameter $U_{d}=8.5$ eV, Hund’s coupling
$J_{d}=1$ eV, and double-counting correction in the atomic limit [20, 21]. All
calculations were performed for the crystallographic unit cell with 120 atoms,
and reciprocal space was sampled by 64 k-points in the symmetry-irreducible
part of the first Brillouin zone. Experimental structural parameters were
used, as further described in Sec. III.4.

##  III Results

###  III.1 Crystal Structure

To the best of our knowledge, synthesis and characterization of BaCuTe2O6 have
not been reported previously. To establish the crystal structure of this
compound, Rietveld refinement of the synchrotron XRD data was performed using
the structural data of SrCuTe2O6 as the starting model [14]. Figure 1 presents
the synchrotron XRD data at two different temperatures along with the Rietveld
fits. The results are listed in Table 1 and suggest close similarity to the Sr
compound. At 80 K, all atomic displacement parameters are reduced with respect
to their room-temperature values, indicating the absence of structural
disorder. The incorporation of Ba has the effect of negative pressure and
increases the Cu–Cu distances compared to SrCuTe2O6 (see also Table 2).

Table 1:  Atomic positions and isotropic atomic displacement parameters ($U_{\rm iso}$, in units of $10^{-2}$ Å2) for BaCuTe2O6 at 80 K (upper lines) and 300 K (lower lines) obtained from the refinement of the synchrotron data. The cubic lattice parameter is $a=12.8296(1)$ Å at 80 K and $a=12.8699(1)$ Å at 300 K. The space group is $P4_{1}32$. All the sites are fully occupied. | Atom | Wyckoff | $x/a$ | $y/a$ | $z/a$ | $U_{\rm iso}$
---|---|---|---|---|---|---
|  | position |  |  |  |
| Ba1 | $8c$ | 0.05757(3) | 0.05757(3) | 0.05757(3) | 0.35(2)
|  |  | 0.05825(4) | 0.05825(4) | 0.05825(4) | 1.29(2)
| Ba2 | $4b$ | 0.375 | 0.625 | 0.125 | 0.41(2)
|  |  | 0.375 | 0.625 | 0.125 | 1.33(3)
| Cu | $12d$ | 0.47429(7) | 0.875 | 0.27571(7) | 0.37(3)
|  |  | 0.47381(9) | 0.875 | 0.27619(9) | 1.17(2)
| Te | $24e$ | 0.33709(4) | 0.91707(4) | 0.06354(4) | 0.39(1)
|  |  | 0.33709(4) | 0.91657(5) | 0.06425(5) | 1.31(1)
| O1 | $24e$ | 0.3763(3) | 0.8256(3) | 0.1734(3) | 0.29(1)
|  |  | 0.3760(4) | 0.8254(4) | 0.1733(4) | 1.01(1)
| O2 | $24e$ | 0.2664(3) | 0.8115(3) | $-0.0121(3)$ | 0.52(1)
|  |  | 0.2673(4) | 0.8117(4) | $-0.0163(4)$ | 1.82(1)
| O3 | $24e$ | 0.2259(3) | 0.9751(3) | 0.1360(4) | 0.89(1)
|  |  | 0.2265(4) | 0.9746(4) | 0.1376(4) | 2.39(1)

Figure 2 shows the crystal structure of BaCuTe2O6. The CuO4 plaquettes are
linked into a 3D network via TeO3 trigonal pyramids. This geometrical
arrangement allows for several interaction topologies, including isolated
triangles formed by the nearest-neighbor interaction $J_{1}$, the hyperkagome
lattice formed by the second-neighbor interaction $J_{2}$, and crossed spin
chains formed by the third-neighbor interaction $J_{3}$ (see also Fig. 10).
This latter interaction is predominant in BaCuTe2O6, as we show in the
following.

Figure 3:  Variation of lattice constant ($a$) and unit cell volume ($V$) with
Zn2+ concentration ($x$), obtained from the Rietveld refinement of the powder
XRD data at room temperature for Ba(Cu1-xZnx)Te2O6. Solid lines are the linear
fits as per the Vegard’s law.

We have also prepared Zn-doped samples that preserve the cubic symmetry up to
the highest doping level of $x\simeq 0.2$. The lattice constant ($a$)
increases almost linearly but weakly with Zn concentration ($x$) (see Fig. 3)
This leads to an overall linear increase in the unit cell volume ($V$). Though
the change is marginal, the increasing trend resembles other Zn-doped Cu2+
compounds [22] and complies with the slightly larger ionic radius of Zn2+ (0.6
Å) compared to Cu2+ (0.57 Å).

###  III.2 Magnetic Susceptibility

Temperature-dependent magnetic susceptibility $\chi(T)$ measured in an applied
magnetic field $\mu_{0}H=1$ T is shown in the upper panel of Fig. 4. It
follows a Curie-Weiss (CW) behavior in the high-temperature regime. A broad
maximum at $T_{\chi}^{\rm max}\simeq 20$ K is the signature of short-range
order, typically observed in low-dimensional antiferromagnetic (AFM) spin
systems. The temperature corresponding to the broad maximum is a measure of
the strength of the dominant AFM exchange interaction [23, 24, 25]. With
decreasing temperature, a sharp peak reminiscent of a transition toward
magnetic long-range-order (LRO) is observed at $T_{\rm N}~{}\simeq 6.1$ K.
When the temperature is lowered further, a small Curie-like upturn in
$\chi(T)$ is seen, which is due to the small amount of extrinsic paramagnetic
impurities or intrinsic defects present in the sample.

Figure 4:  Upper panel: $\chi(T)$ of BaCuTe2O6 measured in an applied field of
$\mu_{0}H=1$ T. The solid red line represents the 1D model fit [Eq. (3)] as
described in the text. Inset: ZFC and FC susceptibilities measured at
$\mu_{0}H=0.01$ T in the low-temperature regime. Lower panel: Inverse
susceptibility $1/\chi$ as a function of temperature. The solid line is the CW
fit using Eq. (2). Inset: Low-temperature $\chi(T)$ of BaCuTe2O6 measured at
different applied fields up to 9 T.

The $\chi(T)$ data in the high-temperature region were fitted by the following
expression:

$\chi(T)=\chi_{0}+\frac{C}{T+\theta_{\rm CW}}.$ (2)

Here $\chi_{0}$ is the temperature-independent susceptibility that includes
contributions of core diamagnetism ($\chi_{\rm core}$) and Van-Vleck
paramagnetism ($\chi_{\rm VV}$). The second term is the CW law with $C$ being
the Curie constant and $\theta_{\rm CW}$ the characteristic Curie-Weiss
temperature. Our fit above 200 K (lower panel of Fig. 4) yields
$\chi_{0}\simeq-1.19\times 10^{-4}$ cm3/mol, $C\simeq 0.41$ cm3 K/mol, and
$\theta_{\rm CW}\simeq 18.97$ K. The value of $C$ corresponds to an effective
moment $\mu_{\rm eff}\simeq 1.82~{}\mu_{\rm B}$, which is slightly higher than
the spin-only value of $\mu_{\rm eff}=g\sqrt{S(S+1)}\mu_{\rm B}\simeq
1.73~{}\mu_{\rm B}$ for Cu2+ ($S=\frac{1}{2}$) and $g=2$. This is consistent
with the $g$-value slightly above 2.0, as typical for Cu2+ compounds [21, 26].
$\chi_{\rm core}$ of BaCuTe2O6 was calculated to be $-1.43\times 10^{-4}$
cm3/mol by adding the $\chi_{\rm core}$ of individual ions Ba2+, Cu2+, Te4+,
and O2- [27]. $\chi_{\rm VV}$ was calculated by subtracting $\chi_{\rm core}$
from $\chi_{0}$ and equals $\sim 2.3\times 10^{-5}$ cm3/mol. This value of
$\chi_{\rm VV}$ is comparable to the other cuprate compounds like Sr2CuO3
[28], Sr2Cu(PO4)2 [29], and PbCu3TeO7 [30]. The value of $\theta_{\rm CW}$ is
positive and comparable in magnitude to $T_{\chi}^{\rm max}$, which implies
that the dominant exchange interaction between the Cu2+ ions is AFM in nature.

Magnetic susceptibility was further analyzed using the expression

$\chi(T)=\chi_{0}+\frac{C_{\rm imp}}{T}+\chi_{\rm spin}(T),$ (3)

where the second term is the Curie law that account for paramagnetic
impurities potentially present in the sample, and $C_{\rm imp}$ gauges the
impurity concentration. The third term $\chi_{\rm spin}(T)$ was taken as the
expression for the spin susceptibility of a uniform 1D Heisenberg
spin-$\frac{1}{2}$ AFM chain given by Johnston et al [23]. This expression is
valid over a wide temperature range ($5\times 10^{-25}\leq k_{\rm B}T/J\leq
5$). The fit above 10 K returned $\chi_{0}\simeq-3.76\times 10^{-5}$ cm3/mol,
$C_{\rm imp}\simeq 0.013$ cm3 K/mol, $g\simeq 2.02$, and $J/k_{\rm B}\simeq
37.0$ K. From the value of $C_{\rm imp}$, the impurity concentration is
estimated to be $\sim 3$%, assuming spin-$\frac{1}{2}$ impurities. Below 10 K,
experimental susceptibility clearly deviates from the fit. Our microscopic
analysis suggests that interchain interactions do not exceed 5 K (Sec. III.4).
Therefore, the deviations are most likely caused by anisotropic terms in the
spin Hamiltonian, such as Dzyloshinskii-Moriya interaction, which is allowed
by symmetry for the intrachain coupling $J_{3}$.

The Sr compound showed two consecutive transitions as a function of
temperature in the magnetic susceptibility and other thermodynamic probes [10,
13, 12]. In contrast, BaCuTe2O6 reveals one transition only (Figs. 4, 5, and
6). Another difference can be seen in very low magnetic fields, where
SrCuTe2O6 displayed no significant deviation between the field-cooled (FC) and
zero-field cooled (ZFC) $\chi(T)$ measurements [10], while BaCuTe2O6 displays
a bifurcation at $T_{\rm N}$ suggestive of a weak spin canting in the
magnetically ordered state (see the inset of the upper panel of Fig. 4).

Figure 5:  Upper panel: Magnetic isotherm ($M$ vs $H$) for BaCuTe2O6 and
SrCuTe2O6 at $T=2$ K. The bend around 3 T in the Sr compound indicates the
transition toward a field-induced state [12]. Inset: Magnified $M$ vs $H$ near
zero field to highlight the minute net moment present in the Ba compound.
Lower panel: Real part of the AC susceptibility ($\chi^{\prime}$) vs $T$
measured on BaCuTe2O6 at different frequencies. The transition temperature is
marked by an arrow.

The magnetization isotherm ($M$ vs $H$) at $T=2$ K measured upto 9 T is shown
in the upper panel of Fig. 5 for both BaCuTe2O6 and SrCuTe2O6. Unlike the Sr
compound, the Ba compound exhibits linear behavior without any indications of
a field-induced transition. In the inset of Fig. 5, we magnified the data near
zero field in order to highlight a tiny remanent magnetization of less than
0.001 $\mu_{B}$/f.u. present in the Ba compound. This indicates a small spin
canting in the antiferromagnetically ordered state.

AC susceptibility as a function of temperature was also measured at different
frequencies. The real part of the AC susceptibility ($\chi^{\prime}$) shows a
peak at $T_{\rm N}$, its position not changing with frequency (lower panel of
Fig. 5). This excludes freezing effects. We also note that the transition
temperature is remarkably insensitive to the applied field (lower panel of
Fig. 4).

###  III.3 Heat Capacity

Figure 6:  Upper panel: $C_{\rm p}$ vs $T$ of BaCuTe2O6 measured in zero
applied field. The dashed and solid lines represent the phononic ($C_{\rm
ph}$) fit [Eq. (4)] and magnetic ($C_{\rm mag}$) parts of $C_{\rm p}(T)$,
respectively. In the inset, $C_{\rm mag}(T)$ is shown in the magnified form.
Lower panel: $C_{\rm mag}/T$ and $S_{\rm mag}$ as a function of $T$ are shown
in the left and right $y$-axes, respectively. The dash-dotted arrow points to
the expected value of magnetic entropy $S_{\rm mag}=R\ln 2$ for Cu2+
($S=\frac{1}{2}$).

Heat capacity $C_{\rm p}$ as a function of temperature measured in zero field
is shown in Fig. 6. With decreasing temperature, it decreases rapidly and then
shows a sharp $\lambda$-type anomaly at $T_{\rm N}\simeq 6.1$ K, affirming the
transition to the magnetically ordered state. In magnetic insulators, $C_{\rm
p}$ has two principal contributions, phononic part and magnetic part that
dominate the signal at higher and lower temperatures, respectively.

To estimate the phononic part of the heat capacity ($C_{\rm ph}$), the raw
data at high temperatures were fitted by a linear combination of four Debye
functions [31, 32],

$C_{\rm ph}(T)=9R\times\sum\limits_{\rm n=1}^{4}c_{\rm
n}\left(\frac{T}{\theta_{\rm Dn}}\right)^{3}\int_{0}^{\frac{\theta_{\rm
Dn}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$ (4)

Here, $R=8.314$ J mol-1 K-1 is the universal gas constant. The coefficients
$c_{\rm n}$ represent the groups of distinct atoms in the crystal, and
$\theta_{\rm Dn}$ are the corresponding Debye temperatures. Since there are
four different atoms with largely different atomic mass, we used four ($n=4$)
Debye functions. The solid dashed curve in Fig. 6 (upper panel) represents the
Debye fit ($C_{\rm ph}$) of the $C_{\rm p}(T)$ data at high temperatures
($T\geq 50$ K) using Eq. (4) and is extrapolated down to low temperatures.

Figure 7:  $C_{\rm mag}/R$ vs $T$ of BaCuTe2O6 in zero magnetic field. The
solid line is the series expansion result for the 1D spin-$\frac{1}{2}$ chain
model from Ref. [23] with the exchange coupling $J/k_{\rm B}\simeq 30$ K.

Magnetic part of the heat capacity ($C_{\rm mag}$) was obtained by subtracting
$C_{\rm ph}$ from the total heat capacity $C_{\rm p}$. The subtraction
procedure was verified by calculating the magnetic entropy $S_{\rm mag}$
through the integration of $C_{\rm mag}(T)/T$ that yields $S_{\rm mag}\simeq
5.87$ J mol-1 K-1 at 80 K (lower panel of Fig. 6). This value is close to the
expected magnetic entropy for spin-$\frac{1}{2}$: $S_{\rm mag}=R\ln 2=5.76$ J
mol-1 K-1. In order to highlight the low-temperature features, we magnified
the low-temperature data in the inset of Fig. 6. $C_{\rm mag}$ shows a clear
broad maximum at $T_{\rm C}^{\rm max}\simeq 13.6$ K representing the short-
range AFM correlations similar to that observed in $\chi(T)$. Both shape and
magnitude of the maximum compare favorably with theoretical results for the
spin-chain model (Fig. 7) [23].

Figure 8:  $C_{\rm p}/T^{2}$ vs $T$ measured under different magnetic fields
from 0 to 9 T. Solid line represents the fit by $C_{\rm p}\propto T^{3}$ in
the low-temperature region, well below $T_{\rm N}$.

Figure 8 shows $C_{\rm p}/T^{2}$ vs $T$ measured in different applied fields
at low temperatures. The value of $T_{\rm N}$ is found to remain unchanged
under external magnetic field up to 9 T similar to that observed in $\chi(T)$.
Below $T_{\rm N}$, $C_{\rm p}(T)$ follows a $T^{3}$ behaviour at all fields,
as expected for a 3D Heisenberg antiferromagnet. This indicates the smallness
(or absence) of an anisotropy gap in the spin-wave spectrum.

###  III.4 Microscopic magnetic model

We now compare and contrast BaCuTe2O6 with its Sr analog. To this end,
magnetic interactions are calculated for the experimental crystal structures
of both compounds determined under exactly the same conditions, by refining
synchrotron XRD data collected at 80 K (the structural parameters for
SrCuTe2O6 are given in the Appendix). This allows us to trace relevant trends
even for weak interchain couplings 111Note that the values reported in this
manuscript for SrCuTe2O6 may be slightly different from those in Ref. [10],
because different structural parameters have been used..

Table 2:  Interatomic distances $d_{i}$ (in Å), hopping parameters $t_{i}$ (in meV), and exchange integrals $J_{i}$ (in K) for SrCuTe2O6 (upper rows) and BaCuTe2O6 (lower rows). The interatomic distances are given according to the experimental crystal structures of both compounds determined at 80 K, only the shortest O$\cdots$O distance is listed. AFM contributions to the exchange couplings are obtained as $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$ with $U_{\rm eff}=4.5$ eV. Total exchange integrals $J_{i}$ are obtained by a GGA+$U$ mapping procedure. The interactions beyond $J_{6}$ are negligible. | $d_{\rm Cu-Cu}$ |  $d_{\rm O\cdots O}$ | $t_{i}$ |  $J_{i}^{\rm AFM}$ | $J_{i}$ |
---|---|---|---|---|---|---
$J_{1}$ | 4.530 | 3.262 | 10 | 1 | 0 | Sr
| 4.736 | 3.412 | 10 | 1 | 0 | Ba
$J_{2}$ | 5.511 | 2.747 | 25 | 7 | $-5$ | Sr
| 5.656 | 2.772 | 46 | 22 | 5 | Ba
$J_{3}$ | 6.276 | 2.773 | $-87$ | 78 | 47 | Sr
| 6.482 | 2.927 | $-73$ | 55 | 34 | Ba
$J_{4}$ | 7.453 | – | 0 | 0 | 0 | Sr
| 7.704 | – | $-2$ | 0 | 0 | Ba
$J_{5}$ | 8.740 | – | 0 | 0 | 0 | Sr
| 9.034 | – | 0 | 0 | 0 | Ba
$J_{6}$ | 8.949 | – | $-20$ | 4 | 2 | Sr
| 9.196 | – | $-18$ | 4 | 2 | Ba
Figure 9:  Superexchange pathways for the interactions $J_{1}-J_{3}$. The
Cu–O–O angles $\alpha$ and $\beta$ control the relative strengths of $J_{2}$
and $J_{3}$. The O$\cdots$O distances are given in Table 2.

Exchange parameters $J_{ij}$ of Eq. (1) are obtained via two complementary
procedures. First, the uncorrelated GGA band structure is mapped onto a tight-
binding model and supplied with an effective Coulomb repulsion $U_{\rm
eff}=4.5$ eV [26, 34] that yields AFM exchange terms as $J_{i}^{\rm
AFM}=4t_{i}^{2}/U_{\rm eff}$, where $t_{i}$ are the hopping parameters. This
way, all relevant exchange couplings, including those between distant Cu
atoms, can be assessed simultaneously. Alternatively, total exchange
couplings, including also ferromagnetic terms, were extracted from GGA+$U$
total energies by a mapping procedure [35, 36].

The results of both methods are listed in Table 2. They suggest that both
compounds are dominated by the third-neighbor interactions $J_{3}$ responsible
for the formation of spin chains. The magnitude of $J_{3}$ decreases upon
replacing Sr with Ba. Microscopically, the coupling $J_{3}$ originates from
the Cu–O$\cdots$O–Cu superexchange controlled by the O$\cdots$O distance (Fig.
9). Such couplings usually occur through the O–O edge of a rigid polyanionic
group, which is rather insensitive to pressure effects or chemical
substitutions [37]. However, in $A$CuTe2O6 the O$\cdots$O contact of $J_{3}$
does not form the edge of the TeO3 pyramid, so the relevant distance can be
quite flexible and increases by about 0.15 Å upon replacing Sr with Ba, thus
reducing $J_{3}$ (Table 2).

The interchain couplings $J_{1}$ and $J_{2}$ follow a similar superexchange
mechanism, but with a largely different interaction geometry (Fig. 9). While
$J_{1}$ is negligible in both compounds, we find a notable increase in
$J_{2}^{\rm AFM}$ and, correspondingly, a change of sign of $J_{2}$ from FM to
AFM upon replacing Sr with Ba. Here, the O$\cdots$O distance is constrained by
the edge of the TeO3 pyramid and changes only weakly, but the linearity of the
pathway gauged by the Cu–O–O angles $\alpha$ and $\beta$ increases: compare
$\alpha_{\rm Sr}=100.6^{\circ}$ and $\beta_{\rm Sr}=155.2^{\circ}$ to
$\alpha_{\rm Ba}=106.1^{\circ}$ and $\beta_{\rm Ba}=159.9^{\circ}$. In the
case of $J_{3}$, $\alpha_{\rm Sr}=\beta_{\rm Sr}=154.5^{\circ}$ and
$\alpha_{\rm Ba}=\beta_{\rm Ba}=153.7^{\circ}$ give rise to much more
“straight” superexchange pathways and cause a much stronger AFM superexchange
than $J_{2}$, despite the longer O$\cdots$O separation. This structural
difference elucidates the $J_{3}\gg|J_{2}|$ regime observed in both Sr and Ba
compounds.

Similar arguments can be invoked to explain the absence of $J_{1}$, where the
O$\cdots$O distance exceeds 3.2 Å (Table 2), while both $\alpha$ and $\beta$
are in the range of $100-110^{\circ}$ and lead to a strong deviation of the
pathway from linearity. Such interaction geometry is prohibitive for the
superexchange (Fig. 9), similar, for example, to fedotovite and other Cu2+
minerals with hexamer clusters [38]. On the other hand, the long-range
interaction $J_{6}$ appears to be non-negligible and comparable in size in
both Sr and Ba compounds (Table 2).

Figure 10:  Frustrated loops formed by the interchain interactions $J_{1}$,
$J_{2}$, and $J_{6}$. Experimental magnetic structure of SrCuTe2O6 [12] is
superimposed. We expect that BaCuTe2O6 has a different spin arrangement
according to the different sign of $J_{2}$ predicted in our work.

We infer that SrCuTe2O6 and BaCuTe2O6 should host similar spin lattices where
$J_{1}$ is nearly absent, whereas $J_{2}$ changes sign from FM for $A={\rm
Sr}$ to AFM for $A={\rm Ba}$. This scenario sheds new light on the magnetic
structure that has been determined for SrCuTe2O6 recently. The neutron
diffraction studies of Refs. [12, 11] revealed that the spins are antiparallel
along the chains, but adopt different directions in different chains.
Altogether, six sublattices with different spin directions are formed. The
spin direction changes not only between every two orthogonal chains, but also
between the two adjacent chains that are parallel to each other [Fig. 10(c)].

In SrCuTe2O6, this strongly non-collinear spin arrangement gives rise to the
coplanar $120^{\circ}$ magnetic order on the triangles formed by $J_{1}$, and
the non-coplanar order on the triangles formed by $J_{2}$ [Fig. 10(a),(b)]. At
first glance, it would indicate that AFM $J_{1}$ should be the leading
interchain coupling, whereas $J_{2}$ is negligible, because any sizable
$J_{2}$ would cause a different type of order on the $J_{2}$-triangles:
collinear ferromagnetic at $J_{2}<0$ and coplanar $120^{\circ}$ at $J_{2}>0$.
Interestingly, we find a sizable $J_{2}$ and neither microscopic indications
nor plausible structural reasons for $J_{1}$ being the leading interchain
coupling in any of the compounds. On the other hand, $J_{6}$ is a plausible
reason for non-collinearity. This coupling connects a spin of one chain to two
adjacent spins in the parallel chain (Fig. 10c). The resulting frustration is
alleviated by arranging the spins of these chains along two orthogonal
directions, similar to the non-collinear magnetic order in Cu2GeO4 [39].

The frustrated nature of $J_{6}$ prevents the spins on the $J_{2}$ triangles
from being parallel, as FM $J_{2}$ would require. However, partial FM
correlations still occur, with the classical interaction energy,

$E_{2}=J_{2}(\mathbf{S}_{1}\mathbf{S}_{2}+\mathbf{S}_{2}\mathbf{S}_{3}+\mathbf{S}_{1}\mathbf{S}_{3})=\tfrac{3}{2}J_{2}S^{2}$
(5)

that allows a sizable energy gain from FM $J_{2}$. This energy gain is in fact
equal to the energy gain expected from the $120^{\circ}$ order on the $J_{1}$
triangle, should the interaction $J_{1}$ be AFM in nature. Therefore, AFM
$J_{1}$ and FM $J_{2}$ are equally instrumental in stabilizing the non-
collinear magnetic order in SrCuTe2O6.

A somewhat similar magnetic structure can be envisaged in BaCuTe2O6. Indeed,
changing the spin directions on a given $J_{2}$ triangle from

$\mathbf{S}_{1}\,\|\,[1\,\bar{1}\,0],\quad\mathbf{S}_{2}\,\|\,[1\,0\,\bar{1}],\quad\mathbf{S}_{3}\,\|\,[0\,\bar{1}\,\bar{1}]$

in SrCuTe2O6 [Fig. 8(b)] to

$\mathbf{S}_{1}\,\|\,[1\,\bar{1}\,0],\quad\mathbf{S}_{2}\,\|\,[\bar{1}\,0\,1],\quad\mathbf{S}_{3}\,\|\,[0\,1\,\bar{1}]$

would result in $E_{2}=-\frac{3}{2}J_{2}S^{2}$ and allow the energy gain from
AFM $J_{2}$. This will lead to a different magnetic symmetry. It would be
interesting to test this prediction experimentally if neutron data of the
quality sufficient to discriminate between different magnetic symmetries could
be obtained for BaCuTe2O6.

###  III.5 Magnetic dilution

Figure 11:  Upper panel: $\chi(T)$ of Ba(Cu1-xZnx)Te2O6 samples measured at
$H=1$ T. Inset: Curie-Weiss temperature $\theta_{\rm CW}$ as a function of
doping concentration $x$ with the solid line representing the linear fit.
Lower panel: $C_{\rm p}/T$ of two end samples ($x=0$ and 0.2) measured in zero
magnetic field.

Zn substitution can have two-fold effect on the magnetism of BaCuTe2O6. The
change in the cubic lattice parameter tunes chemical pressure, whereas the
introduction of non-magnetic Zn2+ ions dilutes the spin lattice. Figure 3
shows that changes in the lattice parameter are minimal, so the dilution
effect should prevail. In the upper panel of Fig. 11, $\chi(T)$ of Zn-doped
samples is plotted after normalizing to one mole of Cu2+ spins. Clearly,
$\chi(T)$ at low temperatures increases systematically with increasing $x$. In
spin systems, when non-magnetic Zn2+ is doped at the magnetic Cu2+ site, free
Cu2+ ions are produced and cause the low-temperature paramagnetic upturn in
$\chi(T)$. Additionally, magnetic dilution may create finite segments of spin
chains that will also give rise to the Curie-like behavior [40]. Thus, our
experimental observations are consistent with the expected dilution effect.

The $\chi(T)$ data at high temperatures (above 200 K) were fitted by the CW
law, Eq. (2). The value of $C$ was found to vary between $\sim 0.41$ and $\sim
0.43$ cm3K/mol-Cu2+, which corresponds to the effective moment $\mu_{\rm
eff}\simeq 1.81-1.85$ $\mu_{\rm B}$. The obtained value of $\theta_{\rm CW}$
is plotted as a function $x$ in the inset of Fig. 11. As the doping level
increases, $\theta_{\rm CW}$ is found to decrease almost linearly from $\sim
19$ K ($x=0$) to $\sim 14$ K ($x=0.2$). This behavior is expected when some of
the superexchange pathways are broken by the dilution, and the overall
interaction energy gauged by $\theta_{\rm CW}$ decreases [22].

Néel temperature should also be reduced by the dilution until long-range
magnetic order disappears completely upon crossing the percolation threshold.
The rate of this reduction and the value of the percolation threshold depend
on the dimensionality of magnetic interactions and can be diagnostic for the
interaction topology. For example, linear decrease in the $T_{N}$ is expected
for quasi-2D antiferromagnets [41], whereas superlinear decrease indicates
further reduction in the dimensionality toward 1D [42] or an additional
magnetic frustration in 2D [43]. Interestingly, BaCuTe2O6 follows neither of
these scenarios. Its Néel temperature is remarkably insensitive to the
dilution. Both the kink in $\chi(T)$ and the $\lambda$-type anomaly in $C_{\rm
p}(T)$ remain around 6.1 K even at $x=0.2$ (Fig. 11). To our knowledge, it is
the only example of a Cu2+-based system, where long-range magnetic order is
essentially unaffected by the Zn2+ substitution. One may understand this
effect as a consequence of magnetic frustration for the interachain couplings,
as discussed in Sec. III.4. Dilution may alleviate the frustration of
interchain couplings and keep $T_{\rm N}$ around the same value even though
the overall energy of magnetic interactions is decreased.

##  IV Conclusions

We have studied the effect of negative pressure on the frustrated network of
the $A$CuTe2O6 compounds. Despite a significant increase in the cubic lattice
parameter and individual Cu–Cu distances, the newly prepared BaCuTe2O6
strongly resembles its Sr sibling and shows a similar quasi-1D magnetic
behavior driven by the dominant third-neighbor couplings $J_{3}$ forming spin
chains. A closer examination of individual magnetic couplings revealed that
the stretching of superexchange pathways caused by negative pressure not only
reduces $J_{3}$, but also increases AFM contribution to the second-neighbor
coupling $J_{2}$ and likely changes its sign. This modification of the
interchain coupling may be the cause of the subtle differences between the Sr
and Ba compounds: the latter lacks the second phase transition at $T_{\rm N2}$
as well as the field-induced transition around 3 T, but reveals minute spin
canting.

We have also shown that the combination of frustrated interchain couplings
$J_{2}$ and $J_{6}$ may stabilize the non-collinear magnetic structure
observed experimentally in SrCuTe2O6 [12, 11]. This scenario allows a
generalized description of the Sr and Ba compounds in terms of a
$J_{2}-J_{3}-J_{6}$ magnetic model, with the nearest-neighbor coupling $J_{1}$
systematically absent in both compounds due to its highly unfavorable
superexchange geometry. The 3D spin-liquid behavior of PbCuTe2O6 requires the
increase in AFM $J_{2}$ with the simultaneous reduction in $J_{3}$ and re-
activation of $J_{1}$ [9]. The structural origin of these rather striking
changes would be an interesting direction for future studies.

Note added: After completing this work, we became aware of an independent
study of BaCuTe2O6 [44, 45]. Neutron diffraction experiments confirm different
magnetic symmetries of SrCuTe2O6 and BaCuTe2O6, in agreement with our
conclusion on the different sign of $J_{2}$ in these compounds.

###### Acknowledgements.

PB, NA, VS, and RN acknowledge SERB, India for financial support bearing
sanction order no. CRG/2019/000960. The work in Augsburg was supported by the
German Research Foundation (DFG) via the Project No. 107745057 (TRR80).

##  Appendix A Crystallographic data for SrCuTe2O6

Here, in Table 3, we present structural parameters for SrCuTe2O6 determined at
80 K using the same experimental setup as in the case of BaCuTe2O6 (Sec. II).
The powder sample of the Sr compound from Ref. [10] was used for the
synchrotron XRD measurement. The results are only marginally different from
the previous reports [14, 12, 11], and facilitate a direct comparison between
the two compounds.

Table 3:  Atomic coordinates and atomic displacement parameters ($U_{\rm iso}$, in units of $10^{-2}$ Å2) for SrCuTe2O6 at 80 K. The lattice parameter is $a=12.4406(1)$ Å, and the space group is $P4_{1}32$.  Atom | Site | $x/a$ | $y/b$ | $z/c$ | $U_{\rm iso}$
---|---|---|---|---|---
Sr1 | $8c$ | 0.05417(4) | 0.05417(4) | 0.05417(4) | 0.43(2)
Sr2 | $4b$ | 0.375 | 0.625 | 0.125 | 0.59(3)
Cu | $12d$ | 0.47634(7) | 0.875 | 0.27366(7) | 0.48(3)
Te | $24e$ | 0.33846(3) | 0.91916(3) | 0.05901(3) | 0.35(1)
O1 | $24e$ | 0.3722(3) | 0.8200(3) | 0.1713(3) | 0.19(9)
O2 | $24e$ | 0.2670(3) | 0.8116(3) | $-0.0227(3)$ | 0.33(9)
O3 | $24e$ | 0.2202(3) | 0.9769(3) | 0.1302(3) | 0.91(9)

## References

  * Savary and Balents [2017] L. Savary and L. Balents, Quantum spin liquids: a review, Rep. Prog. Phys. 80, 016502 (2017).
  * Knolle and Moessner [2019] J. Knolle and R. Moessner, A field guide to spin liquids, Ann. Rev. Condens. Matter Phys. 10, 451 (2019).
  * Broholm  _et al._ [2020] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  * Mendels and Bert [2016] P. Mendels and F. Bert, Quantum kagome frustrated antiferromagnets: One route to quantum spin liquids, C. R. Phys. 17, 455 (2016).
  * Norman [2016] M. R. Norman, Colloquium: Herbertsmithite and the search for the quantum spin liquid, Rev. Mod. Phys. 88, 041002 (2016).
  * Inosov [2018] D. S. Inosov, Quantum magnetism in minerals, Adv. Phys. 67, 149 (2018).
  * Koteswararao  _et al._ [2014] B. Koteswararao, R. Kumar, P. Khuntia, S. Bhowal, S. K. Panda, M. R. Rahman, A. V. Mahajan, I. Dasgupta, M. Baenitz, K. H. Kim, and F. C. Chou, Magnetic properties and heat capacity of the three-dimensional frustrated $S=\frac{1}{2}$ antiferromagnet PbCuTe2O6, Phys. Rev. B 90, 035141 (2014).
  * Khuntia  _et al._ [2016] P. Khuntia, F. Bert, P. Mendels, B. Koteswararao, A. V. Mahajan, M. Baenitz, F. C. Chou, C. Baines, A. Amato, and Y. Furukawa, Spin liquid state in the 3D frustrated antiferromagnet PbCuTe2O6: NMR and muon spin relaxation studies, Phys. Rev. Lett. 116, 107203 (2016).
  * Chillal  _et al._ [2020a] S. Chillal, Y. Iqbal, H. O. Jeschke, J. A. Rodriguez-Rivera, R. Bewley, P. Manuel, D. Khalyavin, P. Steffens, R. Thomale, A. T. M. Nazmul Islam, J. Reuther, and B. Lake, Evidence for a three-dimensional quantum spin liquid in PbCuTe2O6, Nature Comm. 11, 2348 (2020a).
  * Ahmed  _et al._ [2015] N. Ahmed, A. A. Tsirlin, and R. Nath, Multiple magnetic transitions in the spin-$\frac{1}{2}$ chain antiferromagnet ${\mathrm{srcute}}_{2}{\mathrm{o}}_{6}$, Phys. Rev. B 91, 214413 (2015).
  * Saeaun  _et al._ [2020] P. Saeaun, Y. Zhao, P. Piyawongwatthana, T. J. Sato, F. C. Chou, M. Avdeev, G. Gitgeatpong, and K. Matan, Magnetic properties and magnetic structure of the frustrated quasi-one-dimensional antiferromagnet SrCuTe2O6, Phys. Rev. B 102, 134407 (2020).
  * Chillal  _et al._ [2020b] S. Chillal, A. T. M. N. Islam, H. Luetkens, E. Canévet, Y. Skourski, D. Khalyavin, and B. Lake, Magnetic structure of the quantum magnet SrCuTe2O6, Phys. Rev. B 102, 224424 (2020b).
  * Koteswararao  _et al._ [2015] B. Koteswararao, S. K. Panda, R. Kumar, K. Yoo, A. V. Mahajan, I. Dasgupta, B. H. Chen, K. H. Kim, and F. C. Chou, Observation of $S=1/2$ quasi-1D magnetic and magneto-dielectric behavior in a cubic SrCuTe2O6, J. Phys.: Condens. Matter 27, 426001 (2015).
  * Wulff and Müller-Buschbaum [1997] L. Wulff and H. Müller-Buschbaum, Planare CuO4-Polygone und eine einseitig offene Te4+O3-Koordination in SrCuTe2O6, Z. Naturforsch. B 52, 1341 (1997).
  * Momma and Izumi [2011] K. Momma and F. Izumi, VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr. 44, 1272 (2011).
  * Rodríguez-Carvajal [1993] J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Phys. B: Cond. Matt. 192, 55 (1993).
  * Petříček  _et al._ [2014] V. Petříček, M. Dušek, and L. Palatinus, Crystallographic computing system jana2006: general features, Z. Kristallogr. Cryst. Mater. 229, 345 (2014).
  * Koepernik and Eschrig [1999] K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme, Phys. Rev. B 59, 1743 (1999).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Mazurenko  _et al._ [2014] V. V. Mazurenko, M. V. Valentyuk, R. Stern, and A. A. Tsirlin, Nonfrustrated interlayer order and its relevance to the Bose-Einstein condensation of magnons in BaCuSi2O6, Phys. Rev. Lett. 112, 107202 (2014).
  * Nath  _et al._ [2014] R. Nath, K. M. Ranjith, J. Sichelschmidt, M. Baenitz, Y. Skourski, F. Alet, I. Rousochatzakis, and A. A. Tsirlin, Hindered magnetic order from mixed dimensionalities in CuP2O6, Phys. Rev. B 89, 014407 (2014).
  * Ranjith  _et al._ [2015] K. M. Ranjith, M. Majumder, M. Baenitz, A. A. Tsirlin, and R. Nath, Frustrated three-dimensional antiferromagnet Li2CuW2O8: 7Li NMR and the effect of nonmagnetic dilution, Phys. Rev. B 92, 024422 (2015).
  * Johnston  _et al._ [2000] D. C. Johnston, R. K. Kremer, M. Troyer, X. Wang, A. Klümper, S. L. Bud’ko, A. F. Panchula, and P. C. Canfield, Thermodynamics of spin $S=1/2$ antiferromagnetic uniform and alternating-exchange Heisenberg chains, Phys. Rev. B 61, 9558 (2000).
  * Bonner and Fisher [1964] J. C. Bonner and M. E. Fisher, Linear magnetic chains with anisotropic coupling, Phys. Rev. 135, A640 (1964).
  * Eggert  _et al._ [1994] S. Eggert, I. Affleck, and M. Takahashi, Susceptibility of the spin-1/2 Heisenberg antiferromagnetic chain, Phys. Rev. Lett. 73, 332 (1994).
  * Janson  _et al._ [2011] O. Janson, A. A. Tsirlin, J. Sichelschmidt, Y. Skourski, F. Weickert, and H. Rosner, Long-range superexchange in Cu2A2O7 ($A$ = P, As, V) as a key element of the microscopic magnetic model, Phys. Rev. B 83, 094435 (2011).
  * Selwood [1956] P. W. Selwood,  _Magnetochemistry_ , 2nd ed. (Wiley-Interscience, New York, 1956) p. p. 78.
  * Motoyama  _et al._ [1996] N. Motoyama, H. Eisaki, and S. Uchida, Magnetic susceptibility of ideal spin-1/2 Heisenberg antiferromagnetic chain systems, Sr2CuO3 and SrCuO2, Phys. Rev. Lett. 76, 3212 (1996).
  * Nath  _et al._ [2005] R. Nath, A. V. Mahajan, N. Büttgen, C. Kegler, A. Loidl, and J. Bobroff, Study of one-dimensional nature of $S=1/2$ (Sr,Ba)2Cu(PO${}_{4})_{2}$ and BaCuP2O7 via 31P NMR, Phys. Rev. B 71, 174436 (2005).
  * Koteswararao  _et al._ [2013] B. Koteswararao, R. Kumar, J. Chakraborty, B.-G. Jeon, A. V. Mahajan, I. Dasgupta, K. H. Kim, and F. C. Chou, PbCu3TeO7: an staircase kagome lattice with significant intra-plane and inter-plane couplings, J. Phys.: Condens. Matter 25, 336003 (2013).
  * Gopal [1966] E. S. R. Gopal,  _Specific Heats at Low Temperatures_ (Plenum, New York, 1966).
  * Kittel [2005] C. Kittel,  _Introduction to Solid State Physics_ , 8th ed. (Wiley, Hoboken, NJ, 2005).
  * Note [1] Note that the values reported in this manuscript for SrCuTe2O6 may be slightly different from those in Ref. [10], because different structural parameters have been used.
  * Lebernegg  _et al._ [2013] S. Lebernegg, A. A. Tsirlin, O. Janson, and H. Rosner, Spin gap in malachite Cu2(OH)2CO3 and its evolution under pressure, Phys. Rev. B 88, 224406 (2013).
  * Xiang  _et al._ [2011] H. J. Xiang, E. J. Kan, S.-H. Wei, M.-H. Whangbo, and X. G. Gong, Predicting the spin-lattice order of frustrated systems from first principles, Phys. Rev. B 84, 224429 (2011).
  * Tsirlin [2014] A. A. Tsirlin, Spin-chain magnetism and uniform Dzyaloshinsky-Moriya anisotropy in BaV3O8, Phys. Rev. B 89, 014405 (2014).
  * Lebernegg  _et al._ [2011] S. Lebernegg, A. A. Tsirlin, O. Janson, R. Nath, J. Sichelschmidt, Y. Skourski, G. Amthauer, and H. Rosner, Magnetic model for $A_{2}$CuP2O7 ($A$ = Na, Li): One-dimensional versus two-dimensional behavior, Phys. Rev. B 84, 174436 (2011).
  * Nekrasova  _et al._ [2020] D. O. Nekrasova, A. A. Tsirlin, M. Colmont, O. Siidra, H. Vezin, and O. Mentré, Magnetic hexamers interacting in layers in the (Na,K)2Cu3O(SO${}_{4})_{3}$ minerals, Phys. Rev. B 102, 184405 (2020).
  * Badrtdinov  _et al._ [2019] D. I. Badrtdinov, V. V. Mazurenko, and A. A. Tsirlin, Origin of up-up-down-down magnetic order in Cu2GeO4, Phys. Rev. B 100, 214401 (2019).
  * Eggert and Affleck [1995] S. Eggert and I. Affleck, Impurities in $S=1/2$ Heisenberg antiferromagnetic chains: Consequences for neutron scattering and Knight shift, Phys. Rev. Lett. 75, 934 (1995).
  * Chernyshev  _et al._ [2001] A. L. Chernyshev, Y. C. Chen, and A. H. Castro Neto, Long-range order and low-energy spectrum of diluted 2D quantum antiferromagnet, Phys. Rev. Lett. 87, 067209 (2001).
  * Eggert  _et al._ [2002] S. Eggert, I. Affleck, and M. D. P. Horton, Néel order in doped quasi-one-dimensional antiferromagnets, Phys. Rev. Lett. 89, 047202 (2002).
  * Carretta  _et al._ [2011] P. Carretta, G. Prando, S. Sanna, R. De Renzi, C. Decorse, and P. Berthet, Evidence for impurity-induced frustration in La2CuO4, Phys. Rev. B 83, 180411(R) (2011).
  * Samartzis [2020] A. Samartzis,  _Magnetic investigation of effective spin-1/2 magnetic insulators based on 3d and 4f magnetic ions_ , Doctoral thesis, Technische Universität Berlin, Berlin (2020).
  * [45] A. Samartzis, D. Khalyavin, A. T. M. N. Islam, S. Chillal, K. Siemensmeyer, K. Prokes, D. J. Voneshen, A. Senyshyn, and B. Lake, Structural and magnetic properties of the new quantum magnet BaCuTe2O6, arXiv:2102.07490 .


# draft 0

Adel M. Al-Mahdi Mohammad M. Al-Gharabli Mohammad Kafini Department of
Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia.  Shadi Al-Omari

# Stability result for a viscoelastic wave equation in the presence of finite
and infinite memories

Adel M. Al-Mahdi Mohammad M. Al-Gharabli Mohammad Kafini Department of
Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia.  Shadi Al-Omari

###### Abstract

In this paper, we are concerned with the following viscoelastic wave equation

$u_{tt}-\nabla u+\int_{0}^{t}g_{1}(t-s)~{}div(a_{1}(x)\nabla
u(s))~{}ds+\int_{0}^{+\infty}g_{2}(s)~{}div(a_{2}(x)\nabla u(t-s))~{}ds=0,$

in a bounded domain $\Omega$. Under suitable conditions on $a_{1}$ and $a_{2}$
and for a wide class of relaxation functions $g_{1}$ and $g_{2}$. We establish
a general decay result. The proof is based on the multiplier method and makes
use of convex functions and some inequalities. More specifically, we remove
the constraint imposed on the boundedness condition on the initial data
$\nabla u_{0}$. This study generalizes and improves previous literature
outcomes.

Keywords: Stability, Finite and Infinite Memories, Relaxation Functions,
Multiplier Method, Convexity.

AMS Classification.
$35\text{B}40,74\text{D}99,75\text{D}05,93\text{D}15,93\text{D}20$.

##  1 Introduction

Let us consider an $n$-dimensional body occupies with a bounded open set
$\Omega\subseteq\mathbb{R}^{n}(n\geq 1)$ with smooth boundary
$\partial\Omega$. Let $x\rightarrow u(x,t)$ be the position of the material
particle $x$ at time $t$. So that, $\forall x\in\Omega,\forall t>0$, the
corresponding motion equation is

$\begin{array}[]{ll}u_{tt}-\nabla u+\int_{0}^{t}g_{1}(t-s)div(a_{1}(x)\nabla
u(s))ds+\int_{0}^{+\infty}g_{2}(s)div(a_{2}(x)\nabla u(t-s))ds=0,\\\
u(x,t)=0,\\\ u(x,-t)=u_{0}(x,t),u_{t}(x,0)=u_{1}(x).\end{array}$ (1.1)

The functions $g_{1}$ and $g_{2}$ are called the relaxation (kernels)
functions and they are positive non-increasing and defined on
$\mathbb{R}^{+}$. The functions $a_{1}$ and $a_{2}$ are essentially bounded
non-negative defined on $\Omega$. Here, $u_{0}$ and $u_{1}$ are the given
initial data. This model of materials consisting of an elastic part (without
memory) and a viscoelastic, part, where the dissipation given by the memory is
effective.
In this paper, we are concerned with the above viscoelastic wave problem (1.1)
and mainly interested in the asymptotic behavior of the solution $u$ when $t$
tends to infinity. In fact, We prove that the solutions of the corresponding
viscoelastic model decay to zero and no matter how small is the viscoelastic
part of the material. Note that the above model is dissipative, and the
dissipation is given by the memory term only and the memory is effective only
in a part of the body. For materials with memory the stress depends not only
on the present values but also on the entire temporal history of the motion.
Therefore, we have also to prescribe the history of $u$ before 0. Here, we
assume that $u(x,-t)=u_{0}(x,t),\forall x\in\Omega,\forall t>0.$ Let us
mention some other papers related to the problems we address. We start our
literature review with the pioneer work of Dafermos [1], in 1970, where the
author discussed a certain one-dimensional viscoelastic problem, established
some existence results, and then proved that, for smooth monotone decreasing
relaxation functions, the solutions go to zero as $t$ goes to infinity.
However, no rate of decay has been specified. In Dafermos [2], a similar
result, under a ity condition on the kernel, has been established. After that
a great deal of attention has been devoted to the study of viscoelastic
problems and many existence and long-time behavior results have been
established. Hrusa [3] considered a one-dimensional nonlinear viscoelastic
equation of the form

$u_{tt}-cu_{xx}+\int_{0}^{t}m(t-s)\left(\psi(u_{x}(x,s))\right)_{x}~{}ds=f(x,t)$
(1.2)

and proved several global existence results for large data. He also proved an
exponential decay for strong solutions when $m(s)=e^{-s}$ and $\psi$ satisfies
certain conditions. Dassios and Zafiropoulos [4] studied a viscoelastic
problem in $\mathbb{R}^{3}$ and proved a polynomial decay results for
exponentially decaying kernels. After that, a very important contribution by
Rivera was introduced. In 1994, Rivera [5] considered equations for linear
isotropic homogeneous viscoelastic solids of integral type which occupy a
bounded domain or the whole space $\mathbb{R}^{n}$. in the bounded domain
case, and for exponentially decaying memory kernel and regular solutions, he
showed that the sum of the first and the second energy decays exponentially.
For the whole-space case and for exponentially decaying memory kernels, he
showed that the rate of decay of energy is of algebraic type and depends on
the regularity of the solution. this result was later generalized to a
situation, where the kernel is decaying algebraically but not exponentially by
Cabanillas and Rivera [6]. In the paper, the authors considered the case of
bounded domains as well as the case when the material is occupying the entire
space and showed that the decay of solutions is also algebraic, at a rate
which can be determined by the rate of decay of the relaxation function. This
latter result was later improved by Baretto et al. [7], where equations
related to linear viscoelastic plates were treated. Precisely, they showed
that the solution energy decays at the same rate of the relaxation function.
For partially viscoelastic material, Rivera and Salvatierra [8] showed that
the energy decays exponentially, provide the relaxation function decays in a
similar fashion and the dissipation is acting on a part of the domain near to
the boundary. See also, in this direction, the work of Rivera and Oquendo [9].
The uniform decay of solutions for the viscoelastic wave equation

$u_{tt}-k_{0}\nabla u+\int_{0}^{t}div\left[a(x)g(t-\tau)\nabla
u(\tau)\right]d\tau+b(x)h(u_{t})+f(u)=0,$ (1.3)

was investigated by Cavalcanti and Oquendo [10] where they considered the
condition $a(x)+b(x)\geq\delta>0$. They established exponential and polynomial
stability results based on some conditions on $g$ and the linearity of the
function $h$. After that, Guesmiaa and Messaoudib [11] extended the work of
[10] and they establish a general decay result for (1.1) under the same
conditions on $a_{1}$ and $a_{2}$ used in [10] and for some other conditions
for the relaxation functions $g_{1}$ and $g_{2}$, from which the usual
exponential and polynomial decay rates are only special cases. More precisely,
they used the conditions $g_{1}^{\prime}(t)\leq-\xi(t)g_{1}(t),\hskip
10.84006pt\forall t\geq 0,$ and

$\int_{0}^{+\infty}\frac{g_{2}(s)}{G^{-1}(-g_{2}^{\prime}(s))}ds+\sup_{s\in\mathbb{R}_{+}}\frac{g_{2}(s)}{G^{-1}(-g_{2}^{\prime}(s))}<+\infty,$
(1.4)

such that

$G(0)=G^{\prime}(0)=0\hskip 3.61371pt\text{and}\hskip
3.61371pt\lim_{t\rightarrow+\infty}{G^{\prime}(t)}=+\infty,$ (1.5)

where $G:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is an increasing strictly convex
function.
In the present work, we extend the works of [10] and [11]. In fact, we
consider (1.1) and under the same conditions on $a_{1}$ and $a_{2}$ used in
[10] and for a large class of the relaxation functions, we prove a general
decay result. More precisely, we assume that the relaxations functions $g_{1}$
and $g_{2}$ are satisfying

$g_{i}^{\prime}(t)\leq-\xi(t)H_{i}(g_{i}(t)),\hskip 10.84006pt\forall t\geq
0.$

In fact, our result allows a large class of relaxation functions and improves
the decay rates in some earlier papers. The paper is organized as follows. In
Section 2, we present some material needed for our work. Some essential lemmas
are presented and established in Section 3. Section 4 contains the statement
and the proof of our main result. We end our paper by giving some illustrating
examples in Sections 5.

##  2 Preliminaries

In this section, we present some material needed in the proof of our main
result. Through this paper, we use $c$ to denote a positive generic constant.
Now, we start with the following assumptions:
(A1) $g_{i}:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ are differentiable
non-increasing functions such that

$g_{i}(0)>0,~{}~{}i=1,2,~{}~{}~{}~{}~{}1-||a_{1}||_{\infty}\int_{0}^{+\infty}g_{1}(s)ds-||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(s)ds=l>0.$

and there exists $C^{1}$ functions $H_{i}:\mathbb{R}_{+}\to\mathbb{R}_{+}$
which are linear or it is strictly increasing and strictly convex $C^{2}$
function on $(0,r]$ for some $r>0$ with $H_{i}(0)=H_{i}^{\prime}(0)=0$, such
that

$g_{i}^{\prime}(t)\leq-\xi(t)H_{i}(g_{i}(t)),\hskip 10.84006pt\forall t\geq
0,$

where $\xi_{i}:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ are positive
nonincreasing differentiable functions.
(A2) $a_{i}:\bar{\Omega}\longrightarrow\mathbb{R}^{+}$ are in
$C^{1}(\bar{\Omega})$ such that, for positive constant $\delta$ and $a_{0}$
and for $\Gamma_{1},\Gamma_{2}\subset\partial\Omega$ with means
$(\Gamma_{i})>0,~{}i=1,2,~{}\inf_{x\in\bar{\Omega}}(a_{1}(x)+a_{2}(x))\geq\delta$
and

$a_{i}=0~{}~{}~{}\text{or}~{}~{}~{}\inf_{\Gamma_{i}}~{}a_{i}(x)\geq
2a_{0},~{}~{}~{}~{}~{}~{}i=1,2.$

######  Remark 2.1.

If $a_{i}\neq 0,~{}i=1,2,$ there exist neighborhoods $w_{i}$ of
$\Gamma_{i},~{}i=1,2,$ such that

$\inf_{\bar{\Omega\bigcap w_{i}}}a_{i}(x)\geq a_{0}>0,~{}i=1,2.$

As in [10, 11], let $d=\min\\{a_{0},\delta\\}$ and let $\alpha_{i}\in
C^{1}(\bar{\Omega}),~{}i=1,2,$ be such that

$\begin{cases}\begin{array}[]{ll}0\leq\alpha_{i}(x)\leq a_{i}(x)&\\\
\alpha_{i}(x)=0,&\text{if}~{}~{}a_{i}(x)\leq\frac{d}{4}\\\
\alpha_{i}(x)=a_{i}(x),&\text{if}~{}~{}a_{i}(x)\geq\frac{d}{2}.\\\
\end{array}\end{cases}$ (2.1)

######  Lemma 2.2.

The functions $\alpha_{i},~{}i=1,2,$ are not identically zero and satisfy
$\alpha_{1}(x)+\alpha_{2}(x)\geq\frac{d}{2}$.

###### Proof.

(1) For $x\in\Omega\cap w_{i}$, we have $a_{i}(x)\geq a_{0}\geq d$, which
implies, by (2.1), that $\alpha_{i}(x)=a_{i}(x)\geq d.$ Thus $\alpha_{i}$ is
not identically zero.
(2) If $a_{1}(x)\geq\frac{d}{2}$, then $\alpha_{1}(x)=a_{1}(x)$. Consequently
$\alpha_{1}(x)+\alpha_{2}(x)\geq a_{1}(x)\geq\frac{d}{2}$. If
$a_{1}(x)<\frac{d}{2}$, then $a_{2}(x)>\frac{d}{2}$ which implies, by (2.1),
$\alpha_{2}(x)=a_{2}(x)>\frac{d}{2}$. Consequently
$\alpha_{1}(x)+\alpha_{2}(x)>\frac{d}{2}$. This completes the proof. ∎

The existence and uniqueness of the solution of problem (1.1) can be
established by using the Galerkin method. We define the ”modified” energy
functional of the weak solution of problem (1.1), by

$\begin{array}[]{c}E(t)=\frac{1}{2}\int_{\Omega}u_{t}^{2}\partial
x+\frac{1}{2}\int_{\Omega}\left[1-a_{1}(x)\int_{0}^{t}g_{1}(s)ds-
a_{2}(x)\int_{0}^{+\infty}g_{2}(s)ds\right]|\nabla u|^{2}dx\\\
+\frac{1}{2}~{}g_{1}~{}o~{}\nabla u+\frac{1}{2}~{}g_{2}~{}o~{}\nabla
u,\end{array}$ (2.2)

where

$\begin{array}[]{l}g_{1}~{}o~{}\nabla
u=\int_{\Omega}a_{1}(x)\int_{0}^{t}g_{1}(t-s)|\nabla u(t)-\nabla
u(s)|^{2}ds~{}dx\\\ g_{2}~{}o~{}\nabla
u=\int_{\Omega}a_{2}(x)\int_{0}^{+\infty}g_{2}(s)|\nabla u(t)-\nabla
u(t-s)|^{2}ds~{}dx\\\ .\end{array}$ (2.3)

######  Lemma 2.3.

The ”modified” energy functional satisfies, along the solution of problem
(1.1), the following

$E^{\prime}(t)=-\frac{1}{2}~{}g_{1}(t)\int_{\Omega}|\nabla
u(t)|^{2}dx+\frac{1}{2}~{}g^{\prime}_{1}~{}o~{}\nabla
u+\frac{1}{2}~{}g^{\prime}_{2}~{}o~{}\nabla u\leq 0,$ (2.4)

where

$\begin{array}[]{l}g^{\prime}_{1}~{}o~{}\nabla
u=\int_{\Omega}a_{1}(x)\int_{0}^{t}g^{\prime}_{1}(t-s)|\nabla u(t)-\nabla
u(s)|^{2}~{}ds~{}dx\\\ g^{\prime}_{2}~{}o~{}\nabla
u=\int_{\Omega}a_{2}(x)\int_{0}^{+\infty}g^{\prime}_{2}(s)|\nabla u(t)-\nabla
u(t-s)|^{2}~{}ds~{}dx\\\ .\end{array}$ (2.5)

###### Proof.

By multiplying the first equation in problem (1.1) by $u_{t}$ and integrating
over $\Omega$, using integration by parts, hypotheses $(A1)$ and $(A2)$ and
some manipulations as in [6, 12] and others, we obtain (2.4) for regular
solutions. This inequality remains valid for weak solutions by a simple
density argument. ∎

##  3 Technical lemmas

In this section, we introduce some fundamental lemmas. These lemmas will help
us for proving our results. We use our equations and assumptions to prove
those lemmas.

######  Lemma 3.1.

[13]For $i=1,2,$ we have

$\int_{0}^{1}\left(\int_{0}^{t}g_{i}(t-s)\left(\nabla u(s)-\nabla
u(t-s)\right)ds\right)^{2}dx\leq C_{\alpha}(h_{i}\circ\nabla u)(t),$ (3.1)

where, for any $0<\alpha<1$,

$C_{\alpha}=\int_{0}^{t}\frac{g_{i}^{2}(s)}{\alpha
g_{i}(s)-g_{i}^{\prime}(s)}ds\hskip 10.84006pt\text{ and }\hskip
10.84006pth_{i}(t)=\alpha g_{i}(t)-g_{i}^{\prime}(t).$ (3.2)

Furthermore, using the fact that

$\frac{\alpha g_{i}^{2}(s)}{\alpha g_{i}(s)-g_{i}^{\prime}(s)}<g_{i}(s)$

and recalling the Lebesgue dominated convergence theorem, one can deduce that

$\alpha C_{\alpha}=\int_{0}^{\infty}\frac{\alpha g_{i}^{2}(s)}{\alpha
g_{i}(s)-g_{i}^{\prime}(s)}ds\to 0\text{ as }\alpha\to 0.$ (3.3)

######  Lemma 3.2.

There exists a positive constant $M_{1}$ such that

$\displaystyle\int_{\Omega}a_{2}(x)\int_{t}^{+\infty}g_{2}(s)|\nabla
u(t)-\nabla u(t-s)|_{2}^{2}ds\leq M_{1}h_{0}(t),$ (3.4)

where $h_{0}(t)=\int_{0}^{+\infty}g_{2}(t+s)\left(1+||\nabla
u_{0}(s)||^{2}\right)ds$.

###### Proof.

The proof is identical to the one in [14]. Indeed, we have

$\displaystyle\int_{\Omega}a_{2}(x)\int_{t}^{+\infty}g_{2}(s)|\nabla
u(t)-\nabla u(t-s)|_{2}^{2}ds\leq$ (3.5) $\displaystyle
2||a_{2}||_{\infty}||\nabla
u(t)||^{2}\int_{t}^{+\infty}g_{2}(s)ds+2||a_{2}||_{\infty}\int_{t}^{+\infty}g_{2}(s)||\nabla
u(t-s)||^{2}ds$ $\displaystyle\leq 2||a_{2}||_{\infty}\sup_{s\geq 0}||\nabla
u(s)||^{2}\int_{0}^{+\infty}g_{2}(t+s)ds+2||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(t+s)||\nabla
u(-s)||^{2}ds$
$\displaystyle\leq\frac{4||a_{2}||_{\infty}E(s)}{\ell}\int_{0}^{+\infty}g_{2}(t+s)ds+2||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(t+s)||\nabla
u_{0}(s)||^{2}ds$
$\displaystyle\leq\frac{4||a_{2}||_{\infty}E(0)}{\ell}\int_{0}^{+\infty}g_{2}(t+s)ds+2||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(t+s)||\nabla
u_{0}(s)||^{2}ds$ $\displaystyle\leq
M_{1}\int_{0}^{+\infty}g_{2}(t+s)\left(1+||\nabla u_{0}(s)||^{2}\right)ds.$

where
$M_{1}=\max\big{\\{}2||a_{2}||_{\infty},\frac{4||a_{2}||_{\infty}E(0)}{\ell}\big{\\}}.$
∎

######  Lemma 3.3.

[11] Assume that $(A1)$ and $(A2)$ are hold, the functionals

$\psi_{1}(t)=\int_{\Omega}uu_{t}dx,$
$\psi_{2}(t)=-\int_{\Omega}\alpha_{1}(x)u_{t}\int_{0}^{t}g_{1}(t-s)\left(u(t)-u(s)\right)~{}ds~{}dx,$
$\psi_{3}(t)=-\int_{\Omega}\alpha_{2}(x)u_{t}\int_{0}^{+\infty}g_{2}(s)\left(u(t)-u(t-s)\right)~{}ds~{}dx,$

satisfies, along the solution of (1.1) and for any
$\epsilon_{1},\epsilon_{2},\epsilon_{3}>0$ and $\forall t\geq 0$ the following
estimates

$\begin{array}[]{l}\psi_{1}^{\prime}(t)\leq\int_{\Omega}u_{t}^{2}dx-\left[1-\epsilon_{1}-||a_{1}||_{\infty}\int_{0}^{+\infty}g_{1}(s)ds-||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(s)ds\right]\int_{\Omega}|\nabla
u|^{2}dx\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{cC_{\alpha}}{\epsilon_{1}}\left(h_{1}~{}o~{}\nabla
u+h_{2}~{}o~{}\nabla u\right),\end{array}$ (3.6)

$\begin{array}[]{l}\psi^{\prime}_{2}(t)\leq-\left[\int_{0}^{t}g_{1}(s)ds-\epsilon_{2}\right]\int_{\Omega}\alpha_{1}(x)u_{t}^{2}dx+\frac{\epsilon_{3}}{2}\int_{\Omega}|\nabla
u|^{2}dx-\frac{c}{\epsilon_{2}}g^{\prime}_{1}~{}o~{}\nabla u\\\
~{}~{}~{}~{}~{}~{}~{}+\frac{cC_{\alpha}}{\epsilon_{3}}\left(h_{1}~{}o~{}\nabla
u+h_{2}~{}o~{}\nabla u\right),\end{array}$ (3.7)

$\begin{array}[]{l}\psi^{\prime}_{3}(t)\leq-\left[\int_{0}^{+\infty}g_{2}(s)ds-\epsilon_{2}\right]\int_{\Omega}\alpha_{2}(x)u_{t}^{2}dx+\frac{\epsilon_{3}}{2}\int_{\Omega}|\nabla
u|^{2}dx-\frac{c}{\epsilon_{2}}g^{\prime}_{2}~{}o~{}\nabla u\\\
~{}~{}~{}~{}~{}~{}~{}+\frac{cC_{\alpha}}{\epsilon_{3}}\left(h_{1}~{}o~{}\nabla
u+h_{2}~{}o~{}\nabla u\right).\end{array}$ (3.8)

######  Lemma 3.4.

Assume that $(A1)$ and $(A2)$ are hold, then for a suitable choice of
$N,M,m>0$ and for all $t\geq 0$, the functional

$\mathcal{L}(t):=NE(t)+M\psi_{1}(t)+{\psi_{2}}(t)+{\psi_{3}}(t)$

satisfies for any $t>0$

$\mathcal{L}^{\prime}(t)\leq-mE(t)+\frac{1}{4}\bigg{(}g_{1}\circ\nabla
u+g_{2}\circ\nabla u)\bigg{)}(t).$ (3.9)

###### Proof.

Let, for $t_{0}>0$ fixed,
$g_{0}:=\min\\{\int_{0}^{t_{0}}g_{1}(s)ds,\int_{0}^{+\infty}g_{2}(s)ds\\}$.
Then direct differentiation of $\mathcal{L}$ and using (2.4), (4.21), (3.7)
and (3.8) leads to

$\displaystyle\mathcal{L}^{\prime}(t)\leq\bigg{(}\frac{N}{2}-\frac{c}{\varepsilon_{2}}\bigg{)}\bigg{(}(g_{1}^{\prime}\circ\nabla
u)(t)+(g_{2}^{\prime}\circ\nabla u)(t)\bigg{)}$
$\displaystyle-\int_{\Omega}\bigg{[}(g_{0}-\varepsilon_{2})(\alpha_{1}+\alpha_{2})-M\bigg{]}u_{t}^{2}dx$
$\displaystyle\times\bigg{(}\frac{cC_{\alpha}}{\varepsilon_{3}}+\frac{McC_{\alpha}}{\varepsilon_{1}}\bigg{)}\bigg{(}(h_{1}\circ\nabla
u)(t)+(h_{2}\circ\nabla u)(t)\bigg{)}$
$\displaystyle-\bigg{[}(\ell-\varepsilon_{1})M-\varepsilon_{3}\bigg{]}\int_{\Omega}|\nabla
u|^{2}dx,~{}t\geq t_{0}.$

Recalling that $g_{i}^{\prime}=\alpha g_{i}-h_{i}$ where $i=1,2.$ We obtain

$\displaystyle\mathcal{L}^{\prime}(t)$
$\displaystyle\leq\alpha\bigg{(}\frac{N}{2}-\frac{c}{\varepsilon_{2}}\bigg{)}\bigg{(}g_{1}\circ\nabla
u+g_{2}\circ\nabla u\bigg{)}(t)$
$\displaystyle-\bigg{(}\frac{N}{2}-\frac{c}{\varepsilon_{2}}\bigg{)}\bigg{(}h_{1}\circ\nabla
u+h_{2}\circ\nabla u\bigg{)}(t)$
$\displaystyle-\int_{\Omega}\bigg{[}(g_{0}-\varepsilon_{2})(\alpha_{1}+\alpha_{2})-M\bigg{]}u_{t}^{2}dx$
$\displaystyle\times\bigg{(}\frac{cC_{\alpha}}{\varepsilon_{3}}+\frac{McC_{\alpha}}{\varepsilon_{1}}\bigg{)}\bigg{(}h_{1}\circ\nabla
u)(t)+(h_{2}\circ\nabla u)(t)\bigg{)}$
$\displaystyle-\bigg{[}(\ell-\varepsilon_{1})M-\varepsilon_{3}\bigg{]}\int_{\Omega}|\nabla
u|^{2}dx,~{}t\geq t_{0}.$

Simplifying the above estimate, we arrive at

$\displaystyle\mathcal{L}^{\prime}(t)$
$\displaystyle\leq\alpha\bigg{(}\frac{N}{2}-\frac{c}{\varepsilon_{2}}\bigg{)}\bigg{(}g_{1}\circ\nabla
u+g_{2}\circ\nabla u\bigg{)}(t)$
$\displaystyle-\bigg{[}\frac{N}{2}-\frac{c}{\varepsilon_{2}}-C_{\alpha}\big{(}\frac{c}{\varepsilon_{3}}+\frac{Mc}{\varepsilon_{1}}\big{)}\bigg{]}\bigg{(}h_{1}\circ\nabla
u+h_{2}\circ\nabla u\bigg{)}(t)$
$\displaystyle-\int_{\Omega}\bigg{[}(g_{0}-\varepsilon_{2})(\alpha_{1}+\alpha_{2})-M\bigg{]}u_{t}^{2}dx$
$\displaystyle-\bigg{[}(\ell-\varepsilon_{1})M-\varepsilon_{3}\bigg{]}\int_{\Omega}|\nabla
u|^{2}dx,~{}t\geq t_{0}.$

By using the fact that $(\alpha_{1}+\alpha_{2})\geq\frac{d}{2}$ and choosing

$\varepsilon_{1}=\frac{\ell}{2},~{}~{}\varepsilon_{2}=\frac{g_{0}}{2}~{}~{}M=\frac{dg_{0}}{8}^{\prime}~{}~{}\varepsilon_{3}<\frac{d\ell^{2}dg_{0}}{16}$

we get

$(\ell-\varepsilon_{1})M-\varepsilon_{3}\geq 4(1-\ell)$

As a consequence of (3.3), there exists $0<\alpha_{0}<1$ such that if
$\alpha<\alpha_{0}$, then

$\alpha
C_{\alpha}<\frac{1}{8\bigg{[}\frac{c}{\varepsilon_{3}}+\frac{Mc}{\varepsilon_{1}}\bigg{]}}.$
(3.10)

It is clear that (3.10) gives

$C_{\alpha}\bigg{[}\frac{N}{2}-\frac{c}{\varepsilon_{2}}-C_{\alpha}\big{(}\frac{c}{\varepsilon_{3}}+\frac{Mc}{\varepsilon_{1}}\big{)}\bigg{]}<\frac{N}{2}.$
(3.11)

Choosing $\alpha=\frac{1}{2N}<\alpha_{0},$, then for some $m>0$, we have

$\mathcal{L}^{\prime}(t)\leq-mE(t)+\frac{1}{4}\bigg{(}g_{1}\circ\nabla
u+g_{2}\circ\nabla u)\bigg{)}(t).$

Then (3.9) is established. Moreover, one can choose $N$ large enough so that
$\mathcal{L}\sim E$. ∎

######  Lemma 3.5.

For all $t\in\mathbb{R}^{+}$ and fixed positive constants $m_{0}$ , we have
the following estimates

$\int_{0}^{t}E(s)ds<m_{0}\left(1+\int_{0}^{t}h_{0}(s)ds\right).$ (3.12)

where $h_{0}$ is defined in (3.4).

###### Proof.

The proof of this lemma can be established by following the same arguments in
[15, 14]. ∎

######  Lemma 3.6.

If $(A1)-(A2)$ are satisfied, then we have, for all $t>0$ and for $i=1,2$, the
following estimates

$\int_{\Omega}a_{i}(x)\int_{0}^{t}g_{i}(s)|\nabla u(t)-\nabla
u(t-s)|^{2}dsdx\leq\frac{1}{\Lambda_{i}(t)}{H_{i}}^{-1}\left(\frac{\Lambda_{i}(t)\mu_{i}(t)}{\xi_{i}(t)}\right)$
(3.13)

where $\Lambda_{i}(t)=\frac{\lambda_{i}}{1+\int_{0}^{t}h_{0}(s)ds}$;
$\lambda_{i}\in(0,1)$, $H_{i}$ and $\xi_{i}$ are introduced in $(A1)$, and

$\mu_{i}(t):=-\int_{\Omega}a_{i}(x)\int_{0}^{t}g_{i}^{\prime}(s)|\nabla(t)-\nabla(t-s)|^{2}dsdx\leq-
cE^{\prime}(t),$ (3.14)

.

###### Proof.

To establish (3.13), we introduce the following functional

$\chi_{i}(t):=\Lambda_{i}(t)\int_{\Omega}a_{i}(x)\int_{0}^{t}|\nabla
u(t)-\nabla u(t-s)|^{2}dsdx.$ (3.15)

Then, using the fact that $E$ is nonincreasing and (2.2) to get

$\displaystyle\chi_{i}(t)\leq
2\Lambda_{i}(t)\bigg{(}\int_{\Omega}a_{i}(x)\int_{0}^{t}|\nabla
u(t)|^{2}+\int_{\Omega}a_{i}(x)\int_{0}^{t}|\nabla u(t-s)|^{2}dsdx\bigg{)}.$
(3.16) $\displaystyle\hskip
28.90755pt\leq\frac{4\Lambda_{i}(t)||a_{i}||_{\infty}}{\ell}\bigg{(}\int_{0}^{t}\big{(}E(t)+E(t-s)\big{)}ds\bigg{)}$
$\displaystyle\hskip
28.90755pt\leq\frac{8\Lambda_{i}(t)||a_{i}||_{\infty}}{\ell}\int_{0}^{t}E(s)ds$
$\displaystyle\hskip 28.90755pt<+\infty.$

Thus, $\lambda_{i}$ can be chosen so small so that, for all $t>0$,

$\chi_{i}(t)<1.$ (3.17)

Without loss of the generality, for all $t>0$, we assume that $\xi_{i}(t)>0$,
otherwise we get an exponential decay from (3.9). The use of Jensen’s
inequality and using (3.14) and (3.17) gives

$\displaystyle\mu_{1}(t)=\frac{1}{\Lambda_{i}(t)}\int_{0}^{t}\Lambda_{i}(t)(-g_{i}^{\prime}(s))\int_{\Omega}{a_{i}(x)|\nabla(t)-\nabla(t-s)|}^{2}dxds$
(3.18) $\displaystyle\hskip
21.68121pt\geq\frac{1}{\Lambda_{i}(t)}\int_{0}^{t}\Lambda_{i}(t)\xi_{1}(s)H_{i}(g_{i}(s))\int_{\Omega}{a_{i}(x)|\nabla(t)-\nabla(t-s)|}^{2}dxds$
$\displaystyle\hskip
21.68121pt\geq\frac{\xi_{i}(t)}{\Lambda_{i}(t)}\int_{0}^{t}H_{i}(\Lambda_{i}(t)g_{i}(s))\int_{\Omega}{a_{i}(x)|\nabla(t)-\nabla(t-s)|}^{2}dxds$
$\displaystyle\hskip
21.68121pt\geq\frac{\xi_{i}(t)}{\Lambda_{i}(t)}H_{i}\biggl{(}\Lambda_{i}(t)\int_{0}^{t}g_{i}(s)\int_{\Omega}a_{i}(x){|\nabla(t)-\nabla(t-s)|}^{2}dxds\biggr{)}$
$\displaystyle\hskip
21.68121pt=\frac{\xi_{1}(t)}{\Lambda_{i}}\overline{H_{i}}\biggl{(}\Lambda_{i}(t)\int_{0}^{t}g_{i}(s)\int_{\Omega}a_{1}(x){|\nabla
u(t)-\nabla u(t-s)|}^{2}dxds\biggr{)},$

Hence, (3.13) is established. ∎

##  4 Decay result

In this section, we state, prove our main result and provide some example to
illustrate our decay results. Let us start introducing some functions and then
establishing several lemmas needed for the proof of our main result. Now, for
$i=1,2,$ let us take

$\xi=\min\\{\xi_{i}\\},~{}~{}\mu=\max\\{\mu_{i}\\},~{}~{}\lambda=\max\\{\lambda_{i}\\}~{}~{}G=\left(H_{1}^{-1}+H_{2}^{-1}\right)^{-1}.$
(4.1)

As in [14], we introduce the following functions:

$G_{1}(t):=\int_{t}^{r}\frac{1}{sG^{\prime}(s)}ds,~{}~{}G_{5}(t)=G_{1}^{-1}\bigg{(}c_{1}\int_{0}^{t}\xi(s)ds\bigg{)},$
(4.2) $G_{2}(t)=tG^{\prime}(t),\quad G_{3}(t)=t(G^{\prime})^{-1}(t),\quad
G_{4}(t)=G_{3}^{*}(t).$ (4.3)

One can easily verify that $G_{1}$ is decreasing function over $(0,r]$, and
$G_{2},G_{3}$ and $G_{4}$ be convex and increasing functions on $(0,r]$.
Further, we introduce the class $S$ of functions
$\vartheta:\mathbb{R}_{+}\rightarrow(0,r]$ satisfying for fixed
$c_{1},c_{2}>0$,

$\vartheta\in C^{1}(\mathbb{R}_{+}),~{}~{}~{}\vartheta^{\prime}\leq 0,$ (4.4)

and

$c_{2}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]\leq
c_{1}\left(G_{2}\bigg{(}\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}-\frac{G_{2}\left(G_{5}(t)\right)}{\vartheta(t)}\right),$
(4.5)

where $h_{2}$ and $q$ will be defined later in the proof of our main result.

######  Theorem 4.1.

Assume that $(A1)-(A2)$ hold, then there exists a strictly positive constant
$C$ such that the solution of (1.1) satisfies, for all $t\geq 0$,

$E(t)\leq\frac{CG_{5}(t)}{\vartheta(t)\Lambda(t)},$ (4.6)

where $G_{5}$ and $\vartheta$ are defined in (4.2) and (4.4) respectively.

###### Proof.

Using (3.4), (3.9) and (3.13), then for some positive constant $m$, and any
$t\geq 0$, we get

$\displaystyle L^{\prime}(t)\leq-
mE(t)+\frac{c}{\Lambda(t)}G^{-1}\left(\frac{\Lambda(t)\mu(t)}{\xi(t)}\right)+ch_{0}(t).$
(4.7)

Without lose of generality, one can assume that $E(0)>0$. For
$\varepsilon_{0}<r$, let the functional $\mathcal{F}$ defined by

$\mathcal{F}(t):=G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)L(t),$

which satisfies $\mathcal{F}\sim E$. By noting that $G^{\prime\prime}\geq 0,$
$\Lambda^{\prime}\leq 0$ and $E^{\prime}\leq 0$, we get

$\displaystyle\mathcal{F}^{\prime}(t)=\varepsilon_{0}\frac{(qE)^{\prime}(t)}{E(0)}G^{\prime\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)L(t)+G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)L^{\prime}(t)$
(4.8) $\displaystyle\hskip 21.68121pt\leq-
mE(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{c}{\Lambda(t)}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)G^{-1}\left(\frac{\Lambda(t)\mu(t)}{\xi(t)}\right)$
$\displaystyle\hskip
21.68121pt+ch_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right).$

Let $G^{*}$ be the convex conjugate of $G$ in the sense of Young (see [16]),
then

$G^{*}(s)=s(G^{\prime})^{-1}(s)-G\left[(G^{\prime})^{-1}(s)\right],\hskip
7.22743pt\text{if}\hskip 3.61371pts\in(0,G^{\prime}(r)]$ (4.9)

and $G^{*}$ satisfies the following generalized Young inequality

$AB\leq G^{*}(A)+G(B),\hskip 10.84006pt\text{if}\hskip
3.61371ptA\in(0,G^{\prime}(r)],\hskip 3.61371ptB\in(0,r].$ (4.10)

So, with $A=G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
and $B=G^{-1}\left(\frac{\Lambda(t)\mu(t)}{\xi(t)}\right)$ and using
(4.8)-(4.10), we arrive at

$\displaystyle\mathcal{F}^{\prime}(t)\leq-
mE(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{c}{\Lambda(t)}G^{*}\left(G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\right)+c\left(\frac{\mu(t)}{\xi(t)}\right)$
(4.11) $\displaystyle\hskip
21.68121pt+ch_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
$\displaystyle\hskip 21.68121pt\leq-
mE(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+c\varepsilon_{0}\frac{E(t)}{E(0)}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+c\left(\frac{\mu(t)}{\xi(t)}\right)$
$\displaystyle\hskip
21.68121pt+ch_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right).$

So, multiplying (4.11) by $\xi(t)$ and using (3.14) and the fact that
$\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}<r$, gives

$\displaystyle\xi(t)\mathcal{F}^{\prime}(t)\leq-m\xi(t)E(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+c\xi(t)\varepsilon_{0}\frac{E(t)}{E(0)}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
$\displaystyle\hskip
21.68121pt+c\mu(t)\Lambda(t)+c\xi(t)h_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
$\displaystyle\hskip
21.68121pt\leq-\varepsilon_{0}(\frac{mE(0)}{\varepsilon_{0}}-c)\xi(t)\frac{E(t)}{E(0)}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)-cE^{\prime}(t)+c\xi(t)h_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right).$

Consequently, recalling the definition of $G_{2}$ and choosing
$\varepsilon_{0}$ so that $k=(\frac{mE(0)}{\varepsilon_{0}}-c)>0$, we obtain,
for all $t\in\mathbb{R}_{+}$,

$\displaystyle\mathcal{F}_{1}^{\prime}(t)$
$\displaystyle\leq-k\xi(t)\left(\frac{E(t)}{E(0)}\right)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+c\xi(t)h_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
(4.12)
$\displaystyle=-k\frac{\xi(t)}{\Lambda(t)}G_{2}\left(\frac{E(t)\Lambda(t)}{E(0)}\right)+c\xi(t)h_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right),$

where $\mathcal{F}_{1}=\xi\mathcal{F}+cE\sim E$ and satisfies for some
$\alpha_{1},\alpha_{2}>0.$

$\alpha_{1}\mathcal{F}_{1}(t)\leq E(t)\leq\alpha_{2}\mathcal{F}_{1}(t).$
(4.13)

Since $G^{\prime}_{2}(t)=G^{\prime}(t)+tG^{\prime\prime}(t)$ and $G$ is
strictly increasing and strictly convex on $(0,r],$ we find that
$G_{2}^{\prime}(t),G_{2}(t)>0$ on $(0,r].$ Using the general Young inequality
(4.10) on the last term in (4.12) with
$A=G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$ and
$B=[\frac{c}{d}h_{0}(t)]$, we have for $d>0$

$\displaystyle
ch_{0}(t)G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)$
$\displaystyle=\frac{d}{\Lambda(t)}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]\bigg{(}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\bigg{)}$
(4.14)
$\displaystyle\leq\frac{d}{\Lambda(t)}G_{3}\bigg{(}G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\bigg{)}+\frac{d}{\Lambda(t)}G_{3}^{*}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]$
$\displaystyle\leq\frac{d}{\Lambda(t)}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\left(G^{\prime}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\right)+\frac{d}{\Lambda(t)}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]$
$\displaystyle\leq\frac{d}{\Lambda(t)}G_{2}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{d}{\Lambda(t)}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right].$

Now, combining (4.12) and (4.14) and choosing $d$ small enough so that
$k_{0}=(k-d)>0$, we arrive at

$\displaystyle\mathcal{F}_{1}^{\prime}(t)\leq-k\frac{\xi(t)}{\Lambda(t)}G_{2}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{d\xi(t)}{\Lambda(t)}G_{2}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{d\xi(t)}{\Lambda(t)}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]$
(4.15) $\displaystyle\hskip 7.22743pt\leq-
k_{1}\frac{\xi(t)}{\Lambda(t)}G_{2}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)+\frac{d\xi(t)}{\Lambda(t)}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right].$

Using the equivalent property in (4.13) and the increasing of $G_{2}$, we have

$G_{2}\left(\varepsilon_{0}\frac{E(t)\Lambda(t)}{E(0)}\right)\geq
G_{2}\bigg{(}d_{0}\mathcal{F}_{1}(t)\Lambda(t)\bigg{)}.$

Letting $\mathcal{F}_{2}(t):=d_{0}\mathcal{F}_{1}(t)\Lambda(t)$ and recalling
$\Lambda^{\prime}\leq 0$, then we arrive at, for some $c_{1},c_{2}>0$,

$\mathcal{F}_{2}^{\prime}(t)\leq-
c_{1}\xi(t)G_{2}(\mathcal{F}_{2}(t))+c_{2}\xi(t)G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right].$
(4.16)

Since $d_{0}\Lambda(t)$ is nonincreasing. Using the equivalent property
$\mathcal{F}_{1}\sim E$ implies that there exists $b_{0}>0$ such that
$\mathcal{F}_{2}(t)\geq b_{0}E(t)\Lambda(t)$. Let $t\in\mathbb{R}_{+}$ and
$\vartheta(t)$ satisfying (4.4) and (4.5).
If

$b_{0}\Lambda(t)E(t)\leq 2\frac{G_{5}(t)}{\vartheta(t)},$ (4.17)

then, we have

$E(t)\leq\frac{2}{b_{0}}\frac{G_{5}(t)}{\vartheta(t)\Lambda(t)}.$ (4.18)

If

$b_{0}\Lambda(t)E(t)>2\frac{G_{5}(t)}{\vartheta(t)}.$ (4.19)

Then, for any $0\leq s\leq t$, we get

$b_{0}q(s)E(s)>2\frac{G_{5}(t)}{\vartheta(t)},$ (4.20)

since, $\Lambda(t)E(t)$ is nonincreasing function. Therefore, for any $0\leq
s\leq t$, we have

$\mathcal{F}_{2}(s)>2\frac{G_{5}(t)}{\vartheta(t)}.$ (4.21)

Using (4.21), recalling the definition of $G_{2}$, the fact that $G_{2}$ is
convex and $0<\vartheta\leq 1$, we have, for any $0\leq s\leq t$ and
$0<\epsilon_{1}\leq 1$,

$\displaystyle
G_{2}\bigg{(}\epsilon_{1}\vartheta(s)\mathcal{F}_{2}(s)-\epsilon_{1}G_{5}(s)\bigg{)}\leq\epsilon_{1}\vartheta(s)G_{2}\bigg{(}\mathcal{F}_{2}(s)-\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}$
(4.22)
$\displaystyle\leq\epsilon_{1}\vartheta(s)\mathcal{F}_{2}(s)G^{\prime}\bigg{(}\mathcal{F}_{2}(s)-\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}-\epsilon_{1}\vartheta(s)\frac{G_{5}(s)}{\vartheta(s)}G^{\prime}\bigg{(}\mathcal{F}_{2}(s)-\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}$
$\displaystyle\leq\epsilon_{1}\vartheta(s)\mathcal{F}_{2}(s)G^{\prime}\bigg{(}\mathcal{F}_{2}(s)\bigg{)}-\epsilon_{1}\vartheta(s)\frac{G_{5}(s)}{\vartheta(s)}G^{\prime}\bigg{(}\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}.$

Now, we let

$\mathcal{F}_{3}(s)=\epsilon_{1}\vartheta(s)\mathcal{F}_{2}(s)-\epsilon_{1}G_{5}(s),$
(4.23)

where $\epsilon_{1}$ small enough so that $\mathcal{F}_{3}(0)\leq 1$. Then
(4.22) becomes, for any $0\leq s\leq t$,

$\displaystyle
G_{2}\bigg{(}\mathcal{F}_{3}(s)\bigg{)}\leq\epsilon_{1}\vartheta(s)G_{2}\bigg{(}\mathcal{F}_{2}(s)\bigg{)}-\epsilon_{1}\vartheta(s)G_{2}\bigg{(}\frac{G_{5}(s)}{\vartheta(s)}\bigg{)}.$
(4.24)

Further, we have

$\displaystyle\mathcal{F}^{\prime}_{3}(t)=\epsilon_{1}\vartheta^{\prime}(t)\mathcal{F}_{2}(t)+\epsilon_{1}\vartheta(s)\mathcal{F}^{\prime}_{2}(t)-\epsilon_{1}G_{5}^{\prime}(t).$
(4.25)

Since $\vartheta^{\prime}\leq 0$ and using (4.16), then for any $0\leq s\leq
t$, $0<\epsilon_{1}\leq 1$, we obtain

$\displaystyle\mathcal{F}^{\prime}_{3}(t)\leq\epsilon_{1}\vartheta(t)\mathcal{F}^{\prime}_{2}(t)-\epsilon_{1}G_{5}^{\prime}(t)$
(4.26) $\displaystyle\leq-
c_{1}\epsilon_{1}\xi(t)\vartheta(t)G_{2}(\mathcal{F}_{2}(t))+c_{2}\epsilon_{1}\xi(t)\vartheta(s)G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]-\epsilon_{1}G_{5}^{\prime}(t).$

Then, using (4.5) and (4.24), we get

$\displaystyle\mathcal{F}^{\prime}_{3}(t)\leq-
c_{1}\xi(t)G_{2}(\mathcal{F}_{3}(t))+c_{2}\epsilon_{1}\xi(t)\vartheta(t)G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]$
(4.27) $\displaystyle-
c_{1}\epsilon_{1}\xi(t)\vartheta(t)G_{2}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)}\bigg{)}-\epsilon_{1}G_{5}^{\prime}(t).$

From the definition of $G_{1}$ and $G_{5}$, we have

$G_{1}\left(G_{5}(s)\right)=c_{1}\int_{0}^{s}\xi(\tau)d\tau,$

hence,

$G_{5}^{\prime}(s)=-c_{1}\xi(s)G_{2}\left(G_{5}(s)\right).$ (4.28)

Now, we have

$\displaystyle
c_{2}\epsilon_{1}\xi(t)\vartheta(t)G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]-c_{1}\epsilon_{1}\xi(t)\vartheta(t)G_{2}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)}\bigg{)}-\epsilon_{1}G_{5}^{\prime}(t)$
(4.29)
$\displaystyle=c_{2}\epsilon_{1}\xi(t)\vartheta(t)G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]-\epsilon_{1}\xi(t)\vartheta(t)G_{2}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)}\bigg{)}+c\epsilon_{1}\xi(t)G_{2}\left(G_{5}(t)\right)$
$\displaystyle=\epsilon_{1}\xi(t)\vartheta(t)\bigg{(}c_{2}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]-c_{1}G_{2}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)}\bigg{)}+c_{1}\frac{G_{2}\left(G_{5}(t)\right)}{\vartheta(t)}\bigg{)}.$

Then, according to (4.5), we get

$\epsilon_{1}\xi(t)\vartheta(t)\bigg{(}c_{2}G_{4}\left[\frac{c}{d}\Lambda(t)h_{0}(t)\right]-c_{1}G_{2}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)}\bigg{)}-c_{1}\frac{G_{2}\left(G_{5}(t)\right)}{\vartheta(t)}\bigg{)}\leq
0$

Then (4.27) gives

$\displaystyle\mathcal{F}^{\prime}_{3}(t)\leq-
c_{1}\xi(t)G_{2}(\mathcal{F}_{3}(t)).$ (4.30)

Thus from (4.30) and the definition of $G_{1}$ and $G_{2}$ in (4.2) and (4.3),
we obtain

$\bigg{(}G_{1}\left(\mathcal{F}_{3}(t)\right)\bigg{)}^{\prime}\geq
c_{1}\xi(t).$ (4.31)

Integrating (4.31) over $[0,t]$, we get

$G_{1}\left(\mathcal{F}_{3}(t)\right)\geq
c_{1}\int_{0}^{t}\xi(s)ds+G_{1}\left(\mathcal{F}_{3}(0)\right).$ (4.32)

Since $G_{1}$ is decreasing, $\mathcal{F}_{3}(0)\leq 1$ and $G_{1}(1)=0$, then

$\mathcal{F}_{3}(t)\leq
G_{1}^{-1}\bigg{(}c_{1}\int_{0}^{t}\xi(s)ds\bigg{)}=G_{5}(t).$ (4.33)

Recalling that
$\mathcal{F}_{3}(t)=\epsilon_{1}\vartheta(t)\mathcal{F}_{2}(t)-\epsilon_{1}G_{5}(t)$,
we have

$\mathcal{F}_{2}(t)\leq\frac{(1+\epsilon_{1})}{\epsilon_{1}}\frac{G_{5}(t)}{\vartheta(t)},$
(4.34)

Similarly, recall that
$\mathcal{F}_{2}(t):=d_{0}\mathcal{F}_{1}(t)\Lambda(t)$, then

$\mathcal{F}_{1}(t)\leq\frac{(1+\epsilon_{1})}{d_{0}\epsilon_{1}}\frac{G_{5}(t)}{\vartheta(t)\Lambda(t)},$
(4.35)

Since $\mathcal{F}_{1}\sim E$, then for some $b>0$, we have $E(t)\leq
b\mathcal{F}_{1}$; which gives

$E(t)\leq\frac{b(1+\epsilon_{1})}{d_{0}\epsilon_{1}}\frac{G_{5}(t)}{\vartheta(t)\Lambda(t)},$
(4.36)

From (4.18) and (4.36), we obtain the following estimate

$E(t)\leq c_{3}\bigg{(}\frac{G_{5}(t)}{\vartheta(t)\Lambda(t)}\bigg{)},$
(4.37)

where
$c_{3}=\max\\{\frac{2}{b_{0}},\frac{b(1+\epsilon_{1})}{d_{0}\epsilon_{1}}\\}.$
∎

##  5 Examples

Let $g(t)=\frac{a}{(1+t)^{\beta}}$, where $\beta>1$ and $0<a<\beta-1$ so that
$(A1)$ is satisfied. In this case $\xi(t)=\beta a^{\frac{-1}{\beta}}$ and
$G(t)=t^{\frac{\beta+1}{\beta}}$. Then, there exist positive constants
$a_{i}(i=0,...,3)$ depending only on $a,\beta$ such that

$\displaystyle
G_{4}(t)=a_{0}t^{\frac{\beta+1}{\beta}},~{}~{}~{}G_{2}(t)=a_{1}t^{\frac{\beta+1}{\beta}},~{}~{}~{}~{}G_{1}(t)=a_{2}(t^{\frac{-1}{\beta}}-1),~{}~{}G_{5}(t)=(a_{3}t+1)^{-\beta}.$
(5.1)

We will discuss two cases:
Case 1: if

$m_{0}(1+t)^{r}\leq 1+||\nabla u_{0}||^{2}\leq m_{1}(1+t)^{r}$ (5.2)

where $0<r<\beta-1$ and $m_{0},m_{1}>0$, then we have, for some positive
constants $a_{i}(i=4,...,7)$ depending only on $a,\beta,m_{0},m_{1},r$, the
following:

$\displaystyle a_{4}(1+t)^{-\beta+1+r}\leq h_{0}(t)\leq
a_{5}(1+t)^{-\beta+1+r},$ (5.3) $\displaystyle\frac{\lambda}{\Lambda(t)}\geq
a_{6}\left\\{\begin{array}[]{ll}1+\ln(1+t),&\hbox{$\beta-r=2$;}\\\
2,&\hbox{$\beta-r>2$;}\\\ (1+t)^{-\beta+r+2},&\hbox{$1<\beta-r<2$ .}\\\
\end{array}\right.$ (5.4) $\displaystyle\frac{\lambda}{\Lambda(t)}\leq
a_{7}\left\\{\begin{array}[]{ll}1+\ln(1+t),&\hbox{$\beta-r=2$;}\\\
2,&\hbox{$\beta-r>2$;}\\\ (1+t)^{-\beta+r+2},&\hbox{$1<\beta-r<2$ .}\\\
\end{array}\right.$ (5.5)

We notice that condition (4.5) is satisfied if

$(t+1)^{\beta}\Lambda(t)h_{0}(t)\vartheta(t)\leq
a_{8}\bigg{(}1-(\vartheta)^{\frac{1}{\beta}}\bigg{)}^{\frac{\beta}{\beta+1}}.$
(5.6)

where $a_{8}>0$ depending on $a,\beta,c_{1}$ and $c_{2}$. Choosing
$\vartheta(t)$ as the following

$\displaystyle\vartheta(t)=\lambda\left\\{\begin{array}[]{ll}(1+t)^{-p},~{}~{}~{}~{}~{}p=r+1&\hbox{$\beta-r\geq
2$;}\\\ (1+t)^{-p},~{}~{}~{}~{}~{}p=\beta-1,&\hbox{$1<\beta-r<2$ .}\\\
\end{array}\right.$ (5.7)

so that (4.4) is valid. Moreover, using (5.3) and (5.4), we see that (5.6) is
satisfied if $0<\lambda\leq 1$ is small enough, and then (4.5) is satisfied.
Hence (4.6) and (5.5) imply that, for any $t\in\mathbb{R}_{+}$

$\displaystyle E(t)\leq
a_{9}\left\\{\begin{array}[]{ll}\bigg{(}1+\ln(1+t)\bigg{)}(1+t)^{-(\beta-r-1)},&\hbox{$\beta-r=2$;}\\\
(1+t)^{-(\beta-r-1)},&\hbox{$\beta-r>2$;}\\\ (1+t)^{-(\beta-r-1)},&\hbox{
$1<\beta-r<2$ .}\\\ \end{array}\right.$ (5.8)

Thus, the estimate (5.8) gives $\lim_{t\rightarrow+\infty}E(t)=0$.
Case 2: if $m_{0}\leq 1+||\nabla u_{0}||^{2}\leq m_{1}$. That is $r=0$ in
(5.2) as it was assumed in many papers in the literature. So, it clear the
estimate (5.8) gives $\lim_{t\rightarrow+\infty}E(t)=0$ when $r=0$.

## Acknowledgment

The authors thank King Fahd University of Petroleum and Minerals (KFUPM) for
its continuous supports. This work is supported by KFUPM under project #
SB201026.

## Conflict of Interest

The authors declare that there is no conflict of interest regarding the
publication of this paper.

## References

  * [1] C. M. Dafermos, Asymptotic stability in viscoelasticity, Archive for rational mechanics and analysis 37(4) (1970), 297-308.
  * [2] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, Journal of Differential Equations, 7(3) (1970), 554-569.
  * [3] W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data,  SIAM journal on mathematical analysis, 16(1) (1985), 110-134.
  * [4] G. Dassios, Z. Filareti, Equipartition of energy in linearized 3-d viscoelasticity, Quarterly of applied mathematics,  48(4) (1990), 715-730.
  * [5] J. E. M. Rivera, Asymptotic behaviour in linear viscoelasticity, Quarterly of Applied Mathematics,  52(4) (1994), 628-648.
  * [6] J. E. M. Rivera, E. C. Lapa, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels, Communications in mathematical physics, 177(3) (1996), 583-602.
  * [7] J. E. M. Rivera, E. C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, Journal of elasticity,  44(1) (1996), 61-87.
  * [8] J. E. M. Rivera, A. P. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quarterly of Applied Mathematics, 59(3) (2001), 557-578.
  * [9] J. E. M. Rivera, H. P. Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, Journal of elasticity and the physical science of solids 63 (2) (2001), 87-111.
  * [10] M. M. Cavalcanti, H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM journal on control and optimization 42(4) (2003), 1310-1324.
  * [11] A. Guesmia, A. S. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Analysis: Real World Applications 13(1) (2012), 476-485.
  * [12] M. M. Cavalcanti, V. N. D. Cavalcanti, J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electronic Journal of Differential Equations 44(2002), 1-14.
  * [13] M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation Mathematical Methods in the Applied Sciences, 41(1) (2018), 192-204.
  * [14] A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite memory, Mathematical Modelling and Analysis, 25(3), (2020) 351-373.
  * [15] A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, Journal of Mathematical Analysis and Applications, 490(2020), 1-19.
  * [16] V. I. Arnol’d, Mathematical methods of classical mechanics, Springer Science and Business Media, 2013.