A Robba

C Barchasz
email: celine.barchasz@cea.fr

K Bučar

M Petric

M Žitnik

K Kvashnina

G B M Vaughan

R Bouchet

F Alloin

M Kavčič

Fingerprinting mean composition of lithium polysulfide standard solutions by applying high energy resolution fluorescence detected X-ray Absorption Spectroscopy

Keywords: Lithium polysulfide standards; Sulfur K-edge RIXS measurements, sulfur HERFD-XAS spectra, Polysulfide mean chain length

In a lithium/sulfur (Li/S) battery, the reduction of sulfur during discharge involves a particular mechanism, where the active material successively dissolves into the electrolyte to form lithium polysulfide intermediate species (Li 2 S x ), with x being a function of the state of charge.

In this work, sulfur K-edge Resonant Inelastic X-ray Scattering measurements were performed for the characterization of different Li 2 S x polysulfide standard solutions. High Energy Resolution Fluorescence Detected X-ray Absorption Spectroscopy allowed clear separation the pre-edge absorption peak corresponding to terminal sulfur atoms from the main absorption peak due to internal atoms, and to evaluate quantitatively the evolution of the peak area ratio as a function of the polysulfide chain length. Results of this experimental work demonstrate that the normalized area of the pre-edge is a reliable fingerprint of Li 2 S x mean chain length in agreement with recent theoretical predictions. As a perspective, this work confirms that operando HERFD XAS can be used to differentiate mean polysulfide composition, which is key issue in the characterization of Li/S cells.

Lithium/sulfur (Li/S) batteries are one of the most attractive rechargeable energy storage systems, thanks to the low cost and abundance of elemental sulfur, as well as the high theoretical gravimetric energy density attainable. [START_REF] Manthiram | Rechargeable Lithium-Sulfur Batteries[END_REF] This post lithium-ion system usually relies on the use of a lithium metal electrode, an organic electrolyte and a sulfur based cathode where the elemental sulfur (S 8 ) is reduced to produce solid Li 2 S. The electrochemical activity of sulfur (or of the lithiated Li 2 S counterpart) involves a particular cycling mechanism, where the active material successively dissolves into the electrolyte with the formation of lithium polysulfide (PS) intermediate species (Li 2 S x , 2 ≤ x ≤ 8), with x being a function of the stateof-charge, and precipitates at the positive electrode (S 8 and Li 2 S). Thus, the electrochemical processes of Li/S batteries are complex, and still subject to debate. Different discharge mechanisms have already been proposed, involving different lithium PS intermediate species and reaction pathways. [START_REF] Zheng | Investigation of the Li-S Battery Mechanism by Real-Time Monitoring of the Changes of Sulfur and Polysulfide Species during the Discharge and Charge[END_REF][START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF][START_REF] Dominko | Polysulfides Formation in Different Electrolytes from the Perspective of X-ray Absorption Spectroscopy[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF] However, the complex chemical and electrochemical equilibria, as well as the lack of analytical techniques to unambiguously determine the exact composition of Li 2 S x , make the determination of a univocal discharge mechanism very difficult. High-performance liquid chromatography has been applied ex situ, and offers a good selectivity towards each polysulfide. [START_REF] Zheng | Investigation of the Li-S Battery Mechanism by Real-Time Monitoring of the Changes of Sulfur and Polysulfide Species during the Discharge and Charge[END_REF][START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF] However, the detailed understanding of the discharge mechanism requires the use of operando methods, with both good chain length selectivity and time resolution. To achieve this, operando UV-vis and X-ray Absorption (XAS) spectroscopy techniques have been applied to Li/S cells, providing partial information on Li 2 S x intermediate species. [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF][START_REF] Dominko | Polysulfides Formation in Different Electrolytes from the Perspective of X-ray Absorption Spectroscopy[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF][START_REF] Patel | Application of In Operando UV/Vis Spectroscopy in Lithium-Sulfur Batteries[END_REF] In particular, operando X-Ray Absorption Near Edge Structure (XANES) spectroscopy, which is capable of detecting both solid and liquid phases, has been successfully applied in several Li/S battery studies, [START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF][START_REF] Dominko | Analytical Detection of Polysulfides in the Presence of Adsorption Additives by Operando X-ray Absorption Spectroscopy[END_REF][START_REF] Vizintin | The mechanism of Li 2 S activation in lithiumsulfur batteries: Can we avoid the polysulfide formation?[END_REF] giving information relative to the oxidation state of sulfur species including Li 2 S x during the battery cycle.

A linear combination fitting analysis was used in these studies to extract relative amounts of the three main sulfur compounds formed in the battery. Reference spectra used in the fit were either obtained by measuring Li 2 S x standards or Na 2 S x substitutes [START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF] , or in some cases extracted directly from a set of operando spectra [START_REF] Dominko | Analytical Detection of Polysulfides in the Presence of Adsorption Additives by Operando X-ray Absorption Spectroscopy[END_REF] . While the total relative amount of polysulfides produced within the battery could be followed during the cycle, it was only possible to make a rough qualitative estimate of the polysulfide chain length. However, a recent theoretical study [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] suggested a linear relationship between the peak areas of the two main spectral features and the polysulfide chain length, which could be used to differentiate the polysulfides from XANES spectra. In such measurements, the energy resolution of the measured absorption spectra is ultimately limited by the core-hole lifetime broadening and this represents an obstacle in resolving clearly the main features in the Li 2 S x absorption spectrum.

Compared to standard fluorescent mode XAS measurements, energy resolution can be enhanced below the core-hole lifetime broadening using Resonant Inelastic X-ray Scattering (RIXS). In this case, the fluorescence is detected by the emission spectrometer with energy resolution comparable to the resolution of the beamline monochromator. The overall spectral broadening is in this case limited mainly by the experimental resolution (the final state broadening is usually almost negligible), and a significant sharpening effect can be observed in the RIXS spectra. While RIXS in the hard X-ray range is commonly exploited to study 3d transition complexes [START_REF] Glatzel | High resolution 1s core hole X-ray spectroscopy in 3d transition metal complexes-electronic and structural information[END_REF] , the tender X-ray range containing the sulfur absorption edge has been much less explored, mainly due to several technical challenges inherent to measurements in this low energy range.

In our previous work, [START_REF] Kavčič | Operando Resonant Inelastic X-ray Scattering: An Appropriate Tool to Characterize Sulfur in Li-S Batteries[END_REF] operando RIXS measurements at the sulfur K-edge were successfully applied to enhance the sensitivity for lithium polysulfide detection, by exciting the polysulfides resonantly compared to the solid S 8 and Li 2 S counterparts. In the present work, sulfur K-edge RIXS measurements were used for the characterization of different Li 2 S x polysulfides. In order to closely mimic the environment in a Li/S cell, measurements were performed on dissolved lithium PS standard solutions. Enhanced spectral resolution was used to decompose the main spectral features and determine precisely their area ratios, which can be directly compared to theoretical predictions. [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] An example of a full K-L RIXS map is shown in Fig. 1(a) for the Li 2 S 2 equivalent standard solution. The map was built by recording a series of 100 consecutive sulfur K emission spectra while changing the excitation energy across the sulfur K absorption edge in 0.2 eV steps. Due to the high experimental energy resolution, a significant narrowing can be observed in the measured RIXS map, with the first pre-edge resonance due to terminal sulfur atoms well resolved from the main absorption peak corresponding to internal sulfur atoms. By making a horizontal cut through the measured RIXS map at fixed emission energy corresponding to the top of the Kα1 emission line, an absorption like spectrum with superior resolution is obtained. This technique is referred as High Energy Resolution Fluorescence Detected XAS (HERFD-XAS), [START_REF] Hämäläinen | Elimination of the Inner-Shell Lifetime Broadening in X-ray Absorption Spectroscopy[END_REF][START_REF] Glatzel | X-ray Absorption and Emission Spectroscopy[END_REF] and can be used to overcome the intrinsic core-hole lifetime broadening limitation of the XAS spectroscopy. This is demonstrated in Fig. 1(b), comparing the measured Li 2 S 2 sulfur K-edge HERFD-XAS spectrum with the standard XAS spectrum measured as well. Significant improvement in spectral resolution is observed in Fig. 1(b). Compared to standard XAS, the increased resolution of HERFD XAS spectra allows the preedge peak attributed to terminal atoms of lithium PS (~2469 eV) to be clearly resolved from the main peak (~2471 eV) generally attributed to elemental sulfur (S-S bond in S 8 ring). [START_REF] Kavčič | Operando Resonant Inelastic X-ray Scattering: An Appropriate Tool to Characterize Sulfur in Li-S Batteries[END_REF][START_REF] Kavčič | Design and performance of a versatile curved-crystal spectrometer for high-resolution spectroscopy in the tender x-ray range[END_REF]13,[START_REF] Gorlin | Understanding the Charging Mechanism of Lithium-Sulfur Batteries Using Spatially Resolved Operando X-Ray Absorption Spectroscopy[END_REF][START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] Observation of Fig. 2 shows as predicted that the relative area (resp. intensity) of the pre-edge peak increases while the polysulfide chain length decreases. Quantitative analysis can be performed by determining the area (resp. intensity) of the pre-edge peak relative to that of the main one, to better differentiate the polysulfide species. In a recent comprehensive theoretical work of Pascal et al., [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] sulfur XAS spectra of lithium PS dissolved in TEGDME were simulated using first-principles molecular dynamics calculations. Their calculations confirm that the two main absorption features arise from differently coordinated sulfur atoms in the polysulfide molecule. The pre-edge feature is attributed to terminal sulfur atoms associated to lithium atoms at either end of the polysulfide and the main-edge to the internal sulfur atoms. Density functional theory (DFT) calculations revealed increased local electron density (charge) of the terminal atoms compared to the internal ones, leading to reduced binding energy of the core electron and consequently the shift of the corresponding absorption feature to lower energies with respect to those corresponding to internal atoms in the polysulfide chain. This attribution of main absorption features was confirmed also by the work of Kosugi et al., [START_REF] Rühl | Sulfur 1s excitation of S 2 and S 8 : Core-valence-and valence-valence-exchange interaction and geometry-specific transitions[END_REF] who have carried out ab initio calculations on XAS spectrum of S 2 . An almost linear relationship between the normalized peak areas (resp. intensities) of the two main absorption features and the PS chain length was found by Pascal et al. [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] and this is suggested to be used to resolve different PS from XAS spectrum.

In order to check these theoretical predictions, the measured HERFD XAS spectra were fit to the model consisting of both main absorption features followed by an edge. A single Gaussian was used for the pre-edge absorption peak. Due to the high experimental energy resolution, the main absorption peak reveals a slight asymmetry on the high energy side, which can be accounted for by adding two additional arbitrary Gaussians on the high energy tail of the main Gaussian peak. A cumulative Gaussian distribution was used to describe the edge. This model describes well the measured spectra and is used to extract peak area (and intensity) ratios and energy splitting of the two main absorption peaks (Fig. 3 a).

The experimental values can be directly compared to the theoretical predictions of Pascal et al. [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] A comparison of the peak area ratios as a function of PS mean chain length is presented in Fig. 3 b. Very good agreement is obtained for long chain PS Li 2 S x (x≥6), confirming the linear slope predicted by the theory. For shorter chain PS, this slope is less pronounced and the difference between experimental and theoretical values increases. For the shortest Li 2 S 2 PS sample, the experimental spectrum still exhibits both absorption features, whereas only the pre-edge peak due to sulfur external atoms should be observed, as no internal sulfur atoms exist in this shortest polysulfide. This difference is attributed to the fact that the standard solutions do not consist of a pure Li 2 S x species, but of lithium PS mixtures in equilibrium with a mean composition Li 2 S x , with the possible presence of precipitated species. [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF] Indeed, it is well known that lithium PS in solution tend to disproportionate and comproportionate in rapid equilibrium. The presence of chemical equilibria in PS solutions, associated with disproportionation reactions, [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF] is responsible for the formation of multi PS species in one mean Li 2 S x composition . Despite that, the ratio of internal/terminal sulfur atoms in solution, which is the experimental quantity determined by HERFD-XAS, is still conserved and reflects the mean PS chain length corresponding to the nominal x value of the initial Li 2 S X standard solution. On the other hand, such chemical equilibria have an indirect impact on the shorter chains solutions, due to the formation of solid species and/or the incomplete reaction of Li with S 8 . Indeed, Li 2 S (and possibly Li 2 S 2 ) being less soluble than other PS counterparts, [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF] the disproportionation reactions lead to the formation of insoluble products, which precipitate from the solution. [START_REF] Talian | Which Process Limits the Operation of a Li-S System?[END_REF] Due to the precipitation of these species, experimental HERFD spectra correspond to Li 2 S x' solutions, with mean chain length x' larger than the nominal x value of the short chain PS standard solutions. While the absence of solid species due to the high solubility of PS enables the theoretically predicted ratios to be obtained for long PS chains, the precipitation of less soluble short PS chains explains the discrepancy between experimental values and theory, which increases with decreasing the PS chain length. In addition to peak area ratios, the terminal/internal peak splitting have also been extracted from the spectra yielding an average value of 1.46±0.05 eV for all of the standard PS solutions. This is in agreement with the theoretically predicted 1.5 eV value for the longest Li 2 S 8 polysulfide chain. [START_REF] Pascal | X-ray Absorption Spectra of Dissolved Polysulfides in Lithium-Sulfur Batteries from First-Principles[END_REF] According to theoretical calculations, a correlation between peak splitting and partial atomic charge difference is expected so that the splitting should increase to 2.1 eV approaching Li 2 S 3 . However, no significant variation of peak splitting between different measured standards has been observed in the present experiment. Despite significant discrepancies of experimental peak area ratios from the theorical values observed for short chain PS, at least a slight systematic variation of the peak splitting as a function of the PS chain length would be expected, but this is not supported by our experimental data.

To conclude, the increased spectral resolution of HERFD sulfur K-edge absorption spectra of standard PS samples has allowed the clear separation of the pre-edge absorption peak corresponding to terminal sulfur atoms from the main absorption peak due to internal atoms. This allows the quantitative evaluation of the evolution of the peak area ratio as a function of the PS mean chain length. Results of this experimental work generally confirm the theoretical predictions of Pascal et al. that the normalized area of the pre-edge is a reliable fingerprint of Li 2 S x mean chain length. Similar to standard XAS, HERFD XAS can be performed in operando mode, which opens the perspective to use such information to determine and quantify the soluble intermediate species formed in the battery, complementary to X-ray diffraction, which is principally used to quantify the solids formed during the electrochemical process. Therefore, a complete quantitative analysis of sulfur compounds (liquid and solid phases) formed during the electrochemical process is now feasible.

EXPERIMENTAL METHODS

Lithium polysulfide (PS) standard solutions were prepared in an argon-filled glovebox by mixing elemental sulfur (dried, 325 mesh, Alfa Aesar) and metallic lithium in a mixture of tetraethylene glycol (TEGDME, dried, 99%, Adrich) and 1,3-dioxolane (DIOX, anhydrous, 99,8%, Aldrich) with a volume ratio of 1/1. The different lithium PS standard solutions, with equivalent compositions of Li 2 S x (x = 2, 3,…,8), were prepared at 10 mol.L -1 , by using stoichiometric amounts of lithium and sulfur powders. The dispersions reacted at 50°C in a closed tube over a few days to ensure stable composition of the solutions.

For the purpose of RIXS measurements, lithium PS standard solutions were enclosed in a CR2032 coin cell designed for XAS experiments. The coin cells were assembled in an argon filled glovebox, the casing was drilled on one side and airtightness was achieved using a Kapton ® film (7 µm), in order to allow X-rays to reach the sample and collect the fluorescence signal.

Measurements were performed at the ID26 beamline of the European Synchrotron Radiation Facility (ESRF) equipped with a double Si(111) monochromator providing an energy resolution of  0.35 eV at the sulfur K-edge (2472 eV). The photon beam was focused to 50 × 250 m and the incoming flux was ~ 5 × 10 12 photons/s. RIXS spectra were recorded by an in-vacuum Johansson type curved-crystal spectrometer for high-energy resolution X-ray emission spectroscopy in the tender range. [START_REF] Kavčič | Design and performance of a versatile curved-crystal spectrometer for high-resolution spectroscopy in the tender x-ray range[END_REF] The spectrometer operates in the dispersive mode using CCD camera containing 770×1152 pixels with 22.5 m 2 pixel. The first order reflection of the Si(111) analyzer crystal was used for this experiment. At fixed detector position tuned to central Bragg angle corresponding to the S K emission line the spectrometer covered the full bandwidth of  29 eV with energy resolution of  0.4 eV. In addition, a Si photodiode was mounted in the spectrometer chamber to collect also fast XANES spectra, which were used to monitor any possible radiation damage induced by the incident photon beam.

Figure 1 .

 1 Figure 1. RIXS map of Li 2 S 2 10 -1 mol.L -1 standard solution in TEGDME/DIOX 1/1. The

Figure 2 .

 2 Figure 2. Normalized HERFD sulfur K-edge absorption spectra recorded for different lithium PS standard solutions Li 2 S x (x = 2…8) at 10 -1 mol.L -1 in TEGDME/DIOX 1/1. The spectra are

Figure 3 .

 3 Figure 3. (a) HERFD sulfur K-edge absorption spectrum recorded for the Li 2 S 2 standard

AUTHOR INFORMATION

The authors declare no competing financial interests.

ACKNOWLEDGMENT

The authors acknowledge UGA (Université Grenoble Alpes) for Ph.D funding awarded to Alice Robba and thank ESRF for allocation of beamtime and ID26 beamline staff for their assistance during experiment. Work has been supported by the Slovenian Research Program P1-0112.


Mohammad M Al-Gharabli

Aissa Guesmia
email: aissa.guesmia@univ-lorraine.fr

Salim A Messaoudi
email: messaoud@kfupm.edu.sa

WELL-POSEDNESS AND ASYMPTOTIC STABILITY RESULTS FOR A VISCOELASTIC PLATE EQUATION WITH A LOGARITHMIC NONLINEARITY

published or not. The documents may come

Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity

Introduction

In this paper, we deal with the existence and decay of solutions of the following plate problem:

     |u t | ρ u tt + ∆ 2 u + ∆ 2 u tt - t 0 g(t -s)∆ 2 u(s)ds = ku ln |u|,
in Ω × (0, ∞), u(x, t) = ∂u ∂ν (x, t) = 0, in ∂Ω × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, (1.1) where Ω is a bounded domain of R 2 , with a smooth boundary ∂Ω, ν is the unit outer normal to ∂Ω and ρ and k are positive constants. The kernel g is satisfying some conditions to be specified later.

1.1. Problems with a velocity-dependent material density. Cavalcanti et al. [START_REF] Cavalcanti | Existence and uniform decay for nonlinear viscoelastic equation with strong damping[END_REF] considered

     |u t | ρ u tt -∆u -∆u tt + t 0 g(t -s)∆u(s)ds -γ∆u t = 0,
in Ω × (0, ∞), u(x, t) = 0, in ∂Ω × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, (1.2)

1
where Ω is a bounded domain in R n , n ≥ 1, with a smooth boundary ∂Ω, ρ is a positive real number satisfying some conditions and g is a positive exponentially decaying function. They established a global existence result when the constant γ ≥ 0, and an exponential decay result for the case γ > 0. Messaoudi and Tatar [START_REF] Messaoudi | Global existence asymptotic behavior for a non-linear viscoelastic problem[END_REF] extended this decay result to the case where a source term is competing with the viscoelastic and the strong damping. In the absence of the strong damping (γ = 0), Messaoudi and Tatar [START_REF] Messaoudi | Global existence and uniform stability of solutions for a quasilinear viscoelastic problem[END_REF][START_REF] Messaoudi | Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Anal[END_REF] studied (1.2) and showed that the viscoelastic damping is strong enough to drive the system uniformly to rest. Precisely, they showed that the energy of the solution decays exponentially (resp. polynomially) if g decays exponentially (resp. polynomially). Later, Han and Wang [START_REF] Han | General decay of energy for a viscoelastic equation with nonlinear damping[END_REF] considered (1.2) for γ = 0 and with a relaxation function of more general decay type and established, similarly to the work of Messaoudi [START_REF] Messaoudi | General decay of solution energy in a viscoelastic equation with a nonlinear source[END_REF][START_REF] Messaoudi | General decay of solutions of a viscoelastic equation[END_REF], a general decay result in which the usual exponential and the polynomial decay are only special cases. Liu [START_REF] Liu | General decay and blow up of solution for a quasilinear viscoelastic equation with a nonlinear source[END_REF] considered (1.2), for γ = 0, and in the presence of a source term. He established a general decay result similar to the one in [START_REF] Han | General decay of energy for a viscoelastic equation with nonlinear damping[END_REF]. In [START_REF] Liu | General decay rate estimate for a viscoelastic equation with weakly nonlinear timedependent dissipation and source terms[END_REF], Liu studied the problem

     |u t | ρ u tt -∆u -∆u tt + t 0 g(t -s)∆u(s)ds + α(t)h(u t ) = b|u| p-2 u, in Ω × (0, ∞), u(x, t) = 0, in ∂Ω × (0, ∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, (1.3
) and proved, without imposing growth conditions on h, a general decay result which depends on the behavior of g, α and h. Messaoudi and Mustafa [START_REF] Messaoudi | A general stability result for a quasilinear wave equation[END_REF] studied (1.2) for relaxation functions satisfying g (t) ≤ -H(g(t)), (1.4) where H ∈ C 1 (R + ), with H(0) = 0 and H is linear or strictly increasing and strictly convex function C 2 near the origin. They obtained an explicit and general relation between the decay rate for the energy and that of the relaxation function g without imposing restrictive assumptions on the behavior of g at infinity. Recently, Cavalcanti et al. [START_REF] Cavalcanti | Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density[END_REF] considered (1.2), with γ = 0, and a relaxation function satisfying (1.4). In addition, they required lim inf x→0 + {x 2 H (x) -xH (x) + H(x)} ≥ 0 and some other condition and proved that the energy uniformly decays to zero with the rate that is determined from the solutions of the ODE quantifying the behavior of g(t).

Very recently, Messaoudi and Al-Khulaifi [START_REF] Messaoudi | General and optimal decay for a quasilinear viscoelastic equation[END_REF] considered (1.2), with γ = 0, where the relaxation function satisfies (2.2) below and established an optimal and general decay result. 1.2. Plate Problems. Concerning the study of plates, Lagnese [START_REF] Lagnese | Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping[END_REF] studied a viscoelastic plate equation and showed that the energy decays to zero as time goes to infinity by intorducing a dissipative mechanism on the boundary of the system. Rivera et al. [START_REF] Mu Noz Rivera | Decay rates for viscoelastic paltes with memory[END_REF] proved that the first and second order energy, associated with the solutions of the viscoelastic plate equation, decay exponentially provided that the kernel of the memory also decays exponentially. Komornik [START_REF] Komornik | On the nonlinear boundary stabilization of Kirchoff Plates[END_REF] investigated the energy decay of a plate model under weak growth assumptions on the feedback function. Messaoudi [START_REF] Messaoudi | Global existence and nonexistence in a system of Petrovsky[END_REF] studied the following problem:

     u tt + ∆ 2 u + |u t | m-2 u t = |u| p-2 u, in Q T = Ω × (0, T ), u = ∂u ∂ν = 0, on Γ T = ∂Ω × (0, T ), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, (1.5)
and established an existence result and showed that the solution continues to exist globally if m ≥ p, and blows up in finite time if m < p and the initial energy is negative. This result was later improved by Chen and Zhou [START_REF] Zhou | Global nonexistence for a semilinear Petrovsky equation[END_REF].

For boundary damping, Santos and Junior [START_REF] Santos | A boundary condition with memory for Kirchoff plates equations[END_REF] studied the stability of the following problem:

               u tt + ∆ 2 u = 0, in Ω × (0, ∞), u = ∂u ∂ν = 0, on Γ 0 × (0, ∞), -u + t 0 g 1 (t -s)β 1 u(s)ds = 0, on Γ 1 × (0, ∞), ∂u ∂ν + t 0 g 2 (t -s)β 2 u(s)ds = 0, on Γ 2 × (0, ∞), u(0, x) = u 0 (x), u t (0, x) = u 1 (x), in Ω, (1.6)
where

β 1 u = ∆u + (1 -µ)B 1 u and β 2 u = ∂∆u ∂µ + (1 -µ) ∂B 2 u ∂η with B 1 u = 2ν 1 ν 2 u xy -ν 2 1 u yy -ν 2 2 u xx and B 2 u = (ν 1 -ν 2 ) u xy + ν 1 ν 2 (u yy -u xx )
. For more results in this direction, see [START_REF] Benaissa | Energy decay of solutions of a wave equation of φ-Laplacian type with a general weakly nolinear dissipation[END_REF][START_REF] Guesmia | Existence globale et stabilisation interne non linéaire d'un système de Petrovsky[END_REF][START_REF] Han | General Decay Estimate of Energy for the Second Order Evolution Equations with Memory[END_REF][START_REF] Lagnese | Boundary Stabilization of Thin Plates[END_REF][START_REF] Lasiecka | Exponential decay rates for the solutions of Euler-Bernoulli moments only[END_REF].

1.3. Problems with Logarithmic Nonlinearity. The logarithmic nonlinearity is of much interest in physics, since it appears naturally in inflation cosmology and supersymmetric filed theories, quantum mechanics and nuclear physics [START_REF] Barrow | Inflationary models with logarithmic potentials[END_REF][START_REF] Enqvist | Q-balls and baryogenesis in the MSSM[END_REF]. This type of problems has applications in many branches of physics such as nuclear physics, optics and geophysics [START_REF] Bartkowski | One-dimensional Klein-Gordon equation with logarithmic nonlinearities[END_REF][START_REF] Bialynicki-Birula | Wave equations with logarithmic nonlinearities[END_REF][START_REF] Gorka | Logarithmic Klein-Gordon equation[END_REF]. Birula and Mycielski [START_REF] Bialynicki-Birula | Wave equations with logarithmic nonlinearities[END_REF][START_REF] Bialynicki-Birula | Nonlinear wave mechanics[END_REF] studied the following problem:

     u tt -u xx + u -εu ln |u| 2 = 0, in [a, b] × (0, T ), u(a, t) = u(b, t) = 0, in (0, T ), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in [a, b], (1.7)
which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit p → 1 for the p-adic string equation [START_REF] Gorka | Nonlinear equations with infinitely many derivatives[END_REF][START_REF] Vladimirov | The equation of the p-adic open string for the scalar tachyon field[END_REF]. In [START_REF] Cazenave | Equations d'evolution avec non-linearite logarithmique[END_REF], Cazenave and Haraux considered

u tt -∆u = u ln |u| k , in R 3 (1.8)
and established the existence and uniqueness of the solution for the Cauchy problem. Gorka [START_REF] Gorka | Logarithmic Klein-Gordon equation[END_REF] used some compactness arguments and obtained the global existence of weak solutions, for all

(u 0 , u 1 ) ∈ H 1 0 ([a, b]) × L 2 ([a, b]
), to the initial-boundary value problem (1.8) in the one-dimensional case. Bartkowski and Gorka [START_REF] Bartkowski | One-dimensional Klein-Gordon equation with logarithmic nonlinearities[END_REF] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for equation (1.8). Hiramatsu et al. [START_REF] Hiramatsu | Numerical study of Q-ball formation in gravity mediation[END_REF] introduced the following equation:

u tt -∆u + u + u t + |u| 2 u = u ln |u| (1.9)
to study the dynamics of Q-ball in theoretical physics and presented a numerical study. However, there was no theoretical analysis for the problem. In [START_REF] Han | Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics[END_REF], Han proved the global existence of weak solutions, for all (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω), to the initial boundary value problem (1.9) in R 3 .

In this paper, we are concerned with the well-posedness and stability of the plate problem (1.1) with kernels g having an arbitrary growth at infinity (condition (2.2) below). The obtained stability results improve and generalize many results in the literature.

This paper is organized as follows. In section 2, we present some notations and material needed for our work. In section 3, we establish the local existence of the solutions of the problem. The global existence and the decay results are presented in section 4 and section 5, respectively.

Preliminaries

In this section, we present some material needed for the proof of our results. We use the standard Lebesgue space L 2 (Ω) and Sobolev space H 2 0 (Ω) with their usual scalar products and norms. Throughout this paper, c is used to denote a generic positive constant.

We consider the following hypotheses: (A1) g : R + → R + is a C 1 -nonincreasing function satisfying

g(0) > 0, 1 - ∞ 0 g(s)ds = > 0. ( 2

.1)

(A2) There exist a nonincreasing differentiable function ξ : R + → R + , with ξ(0) > 0, and a constant 1 ≤ p < 3 2 such that g (t) ≤ -ξ(t)g p (t), ∀t ∈ R + .

(2.2)

(A3) The constant k in (1.1) is such that 0 < k < k 0 = 2π e 3 c p , (2.3)
where c p is the smallest positive number satisfying

∇u 2 2 ≤ c p ∆u 2 2 , ∀u ∈ H 2 0 (Ω), where . 2 = . L 2 (Ω) .
The energy functional associated with problem (1.1) is

E(t) = 1 ρ + 2 u t ρ+2 ρ+2 + 1 2 1 - t 0 g(s)ds ∆u 2 2 + ∆u t 2 2 -k Ω u 2 ln |u|dx + k 4 u 2 2 + 1 2 (go∆u), (2.4)
where

(go∆u)(t) = t 0 g(t -s) ∆u(s) -∆u(t) 2 2 ds.
Direct differentiation of (2.4), using (1.1), leads to

E (t) = 1 2 (g o∆u)(t) - 1 2 g(t) ∆u 2 2 ≤ 1 2 (g o∆u)(t) ≤ 0. ( 2
: [0, T ] → [1, ∞) satisfies w(t) ≤ C 1 + t 0 γ(s)w(s) ln (w(s))ds , ∀t ∈ [0, T ]. (2.8)
Then

w(t) ≤ C exp C t 0 γ(s)ds , ∀t ∈ [0, T ].
(2.9)

Lemma 2.4. Let 0 ∈ (0, 1). Then there exists d ε0 > 0 such that s| ln s| ≤ s 2 + d 0 s 1-0 , ∀s > 0.

(2.10)

Proof. Let r(s) = s 0 (| ln s| -s). Notice that r is continuous on (0, ∞) and its limit at 0 + is 0 + , and its limit at ∞ is -∞. Then r has a maximum d 0 on (0, ∞), so (2.10) holds.

Local existence

In this section, we state and prove the local existence result for problem (1.1).

Definition 3.1. Let T > 0. A function u ∈ C 1 ([0, T ], H 2 0 (Ω)) is called a weak solution of (1.1) on [0, T ] if      Ω |u t | ρ u tt (x, t)w(x)dx + Ω ∆u(x, t)∆w(x)dx + Ω ∆u tt ∆wdx -Ω ∆w(x) t 0 g(t -s)∆u(s)dsdx = k Ω w(x)u(x, t) ln |u(x, t)|dx, ∀w ∈ H 2 0 (Ω), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x). (3.1) Theorem 3.2. Assume that (A1) -(A3) hold and let (u 0 , u 1 ) ∈ H 2 0 (Ω) × H 2 0 (Ω). Then problem (1.1) has a weak solution on [0, T ].
Proof. To establish the existence of a solution to problem (1.1), we use the Faedo-Galerkin approximations. Let {w j } ∞ j=1 be an orthogonal basis of the "separable" space H 2 0 (Ω). Let V m = span{w 1 , w 2 , ..., w m } and let the projections of the initial data on the finite dimensional subspace V m be given by

u m 0 (x) = m j=1 a j w j (x), u m 1 (x) = m j=1 b j w j (x),
where

u m 0 → u 0 in H 2 0 (Ω) and u m 1 → u in H 2 0 (Ω), as m → ∞. (3.
2) We search for an approximate solution

u m (x, t) = m j=1 h m j (t)w j (x) of the approximate problem in V m            Ω |u m t | ρ u m tt w + ∆u m ∆w + ∆u m tt ∆w - t 0 g(t -s)∆u m (s)∆wds dx = k Ω wu m ln |u m |dx, ∀w ∈ V m , u m (0) = u m 0 = m j=1 (u 0 , w j )w j , u m t (0) = u m 1 = m j=1 (u 1 , w j )w j . (3.3)
This leads to a system of ODEs for unknown functions h m j (t). Based on standard existence theory for ODE, one can obtain functions

h j : [0, t m ) → R, j = 1, 2, ..., m,
which satisfy (3.3) in a maximal interval [0, t m ), t m ∈ (0, T ]. Next, we show that t m = T and that the local solution is uniformly bounded independently of m and t. For this purpose, let w = u m t in (3.3) and integrate by parts to obtain

d dt E m (t) ≤ 1 2 (g o∆u m ) ≤ 0, (3.4)
where

E m (t) = 1 ρ + 2 u m t ρ+2 ρ+2 + 1 2 ∆u m t 2 2 + 1 - t 0 g(s)ds ∆u m 2 2 + (go∆u m )(t) + k 4 u m 2 2 - k 2 Ω |u m | 2 ln |u m |dx. (3.5) From (3.4), we have E m (t) ≤ E m (0), ∀t ≥ 0.
The last inequality together with the Logarithmic Sobolev inequality leads to

u m t ρ+2 ρ+2 + ∆u m t 2 2 + - ka 2 c p 2π ∆u m 2 2 + k 2 + k (1 + ln a) u m 2 2 + go∆u m ≤ C + u m 2 2 ln u m 2 2 , (3.6) where C = 2E m (0). Choosing e -3 2 < a < 2π kc p (3.7)
will make

- ka 2 c p 2π > 0 and k 2 + k (1 + ln a) > 0.
This selection is possible thanks to (A3). So, we get

u m t ρ+2 ρ+2 + ∆u m t 2 2 + ∆u m 2 2 + u m 2 2 + go∆u m ≤ c 1 + u m 2 2 ln u m 2 2 . (3.8) Let us note that u m (., t) = u m (., 0) + t 0 ∂u m ∂s (., s)ds.
Then, using Cauchy-Schwarz' inequality, we get

u m (t) 2 2 ≤ 2 u m (0) 2 2 + 2 t 0 ∂u m ∂s (s)ds 2 2 ≤ 2 u m (0) 2 2 + 2T t 0 u m t (s) 2 2 ds, (3.9)
hence, inequality (3.8) gives

u m 2 2 ≤ 2 u m (0) 2 2 + 2T c 1 + t 0 u m 2 2 ln u m 2 2 ds . (3.10) If we put C 1 = max {2T c, 2 u(0) 2 2 }, (3.10) leads to u m 2 2 ≤ 2C 1 1 + t 0 u m 2 2 ln u m 2 2 ds .
Because C 1 ≥ 0, we get

u m 2 2 ≤ 2C 1 1 + t 0 C 1 + u m 2 2 ln C 1 + u m 2 2 ds .
Applying the Logarithmic Gronwall inequality to the last inequality, we obtain the following estimate:

u m 2 2 ≤ 2C 1 e 2C1T = C 2 . Hence, from inequality (3.8) it follows that (go∆u m )(t) + u m t ρ+2 ρ+2 + ∆u m t 2 2 + ∆u m 2 2 + u m 2 2 ≤ c(1 + C 2 ln C 2 ) = C 3 .
This implies sup t∈(0,tm)

(go∆u m )(t) + u m t ρ+2 ρ+2 + ∆u m t 2 2 + ∆u m 2 2 + u m 2 2 ≤ C 3 . (3.11)
So, the approximate solution is uniformly bounded independent of m and t. Therefore, we can extend t m to T .

Substituting w = u m tt in (3.3) and using Young's and Cauchy-Schwarz' inequalities, we obtain

Ω |u m t | ρ |u m tt | 2 dx + ∆u m tt 2 2 = - Ω ∆u m ∆u m tt dx + Ω t 0 g(t -s)∆u m (s)∆u m tt (t)dsdx + k Ω u m tt u m ln |u m |dx ≤ δ ∆u m tt 2 2 + 1 4δ t 0 g(t -s) ∆u m (s) 2 ds 2 + δ ∆u m tt + 1 4δ ∆u m (t) 2 + k Ω u m tt u m ln |u m |dx.
(3.12) To estimate the last term in the right-hand side of (3.12), we apply (2.10) with ε 0 = 1 2 and use repeatedly Young's, Cauchy-Schwarz' and the embedding inequalities as follows

k Ω u m tt u m ln |u m |dx ≤ c Ω u m tt |u m | 2 + d 2 √ u m dx ≤ c δ Ω |u m tt | 2 dx + 1 4δ Ω |u m | 2 + d 2 √ u m 2 dx ≤ cδ ∆u m tt 2 2 + c 4δ Ω |u m | 4 dx + Ω |u m |dx ≤ cδ ∆u m tt 2 2 + c 4δ ∆u m 4 2 + u m 2 . (3.13)
Combining (3.12) and (3.13) to have

Ω |u m t | ρ |u m tt | 2 dx + (1 -cδ) ∆u m tt 2 2 ≤ 1 4δ t 0 g(t -s) ∆u m (s) 2 ds 2 + 1 4δ ∆u m 2 + c δ ∆u m 4 2 + u m 2 2 .
(3.14)

Integrate the last inequality on (0, T ) and use (2.1) and (3.11), we obtain

T 0 Ω |u m t | ρ |u m tt | 2 dxdt + (1 -cδ) T 0 ∆u m tt 2 2 dt ≤ c δ T 0 (go∆u m )(t) + ∆u m 2 2 + ∆u m 4 2 + u m 2 2 dt. (3.15)
From the last inequality, choosing δ > 0 small enough and using (3.11), we get the following, for some positive constant C 4 not depending neither on m nor on t:

     (u m ) is uniformly bounded in L ∞ (0, T ; H 2 0 (Ω)), (u m t ) is uniformly bounded in L ∞ (0, T ; L ρ+2 (Ω)) ∩ L ∞ (0, T ; H 2 0 (Ω)), (u m tt ) is uniformly bounded in L 2 (0, T ; H 2 0 (Ω)), (3.17)
which implies that there exists a subsequence of (u m ) (still denoted by (u m )), such that

               u m u weakly * in L ∞ (0, T ; H 2 0 (Ω)), u m t u t weakly * in L ∞ (0, T ; L ρ+2 (Ω)) ∩ L ∞ (0, T ; H 2 0 (Ω)), u m u weakly in L 2 (0, T ; H 2 0 (Ω)), u m t u t weakly in L 2 (0, T ; L ρ+2 (Ω)) ∩ L 2 (0, T ; H 2 0 (Ω)), u m tt w u tt in L 2 (0, T ; H 2 0 (Ω)). (3.18)
Analysis of the non-linear terms

(1) Term u m ln |u m |: using (3.17), we have (u m ) is bounded in L ∞ (0, T ; H 2 0 (Ω)) which implies, using the embedding of

H 2 0 (Ω) in L ∞ (Ω) (Ω ⊂ R 2 ), the boundedness of (u m ) in L 2 (Ω × (0, T )). Similarly; (u m t ) is bounded in L 2 (Ω × (0, T )).
Then, making use of Aubin-Lions' theorem, we find, up to a subsequence, that

u m → u strongly in L 2 (Ω × (0, T )) and u m → u a.e. in Ω × (0, T ).
Since the maps s → ks ln |s| is continuous, we have the following convergence:

ku m ln |u m | → ku ln |u| a.e. in Ω × (0, T ). (3.19)
Using the embedding of (3.20)

H 2 0 (Ω) in L ∞ (Ω) (Ω ⊂ R 2 ), it is clear that k(u m ln |u m |) is bounded in L ∞ (Ω × (0, T )). Next,
(2) Term |u m t | ρ u m t : using (3.16), we have (u m t ) is uniformly bounded in L ∞ (0, T ; H 2 0 (Ω)) which implies the boundedness of (u m t ) in L ∞ (Ω × (0, T )), and so in L 2 (Ω × (0, T )). Using (3.16), we see that (u m tt ) is bounded in L 2 ((0, T ); H 2 0 (Ω)) which implies that (u m tt ) is bounded in L 2 (Ω × (0, T )). Now, using Aubin-Lions theorem, there exists a subsequence, still denoted by (u m t ), such that u m t → u t strongly in L 2 (0, T ; L 2 (Ω)) and

|u m t | ρ u m t → |u t | ρ u t a.e. in Ω × (0, T ). (3.21)
Using (3.11) and the embedding theorems, we have Now, we integrate (3.3) over (0, t) to obtain, for every w ∈ V m , This ends the proof of Theorem 3.2.

|u m t | ρ u t 2 L 2 (0,T ;L 2 (Ω)) = T 0 u m t 2(ρ+1) 2(ρ+1) dt ≤ c T 0 ∆u m t (t) 2(ρ+1) 2 dt ≤ cT C ρ+1 3 , (3.22
1 ρ + 1 Ω |u m t | ρ u m t wdxds - 1 ρ + 1 Ω |u m 1 |u m 1 wdx + t 0 Ω ∆u m (s)∆wdxds + Ω ∆u m t ∆wdxds - Ω ∆u m 1 ∆wdx - Ω t 0 τ 0 g(τ -s)∆u m (s) ∆wdsdτ dx = k t 0 Ω wu m (s)
1 ρ + 1 t 0 Ω |u s | ρ u s wdxds = 1 ρ + 1 Ω |u m 1 |u m 1 wdx - t 0 Ω ∆u(s)∆wdxds - Ω ∆u t ∆wdxds + Ω ∆u m 1 ∆wdx + Ω t 0 τ 0 g(τ -s)∆u(s) ∆wdsdτ dx + k t 0 Ω wu(s) ln |u(s)|dxds, (3.25

Global Existence

In this section, we state and prove a global existence result under smallness conditions on the initial data (u 0 , u 1 ). For this purpose, we introduce the following functionals:

J(t) = 1 2 1 - t 0 g(s)ds ∆u 2 2 + ∆u t 2 2 + go∆u -k Ω u 2 ln |u|dx + k 4 u 2 2 (4.1)
and

I(t) = 1 - t 0 g(s)ds ∆u 2 2 + ∆u t 2 2 + go∆u -3k Ω u 2 ln |u|dx. (4.2)
Lemma 4.1. The following inequalities hold:

-kd 0 |Ω|c 3 * ∆u 3 2 2 ≤ k Ω u 2 ln |u|dx ≤ kc 3 * ∆u 3 2 , ∀u ∈ H 2 0 (Ω), (4.3)
where d 0 = sup 0<s<1 √ s| ln s| = 2 e , |Ω| is the Lesbegue measure of Ω and c * is the smallest embedding constant

Ω |u| 3 dx 1 3 ≤ c * ∆u 2 , ∀u ∈ H 2 0 (Ω) (4.4)
(c * exists thanks to the embedding of

H 2 0 (Ω) in L ∞ (Ω) and Ω ⊂ R 2 ). Proof. Let Ω 1 = {x ∈ Ω : |u(x)| > 1} and Ω 2 = {x ∈ Ω : |u(x)| ≤ 1}.
So, using (4.4), we have

k Ω u 2 ln |u|dx = k Ω2 u 2 ln |u|dx + k Ω1 u 2 ln |u|dx ≤ k Ω1 u 2 ln |u|dx ≤ k Ω1 |u| 3 dx ≤ k Ω |u| 3 dx ≤ kc 3 * ∆u 3 2 .
On the other hand, using Hölder's inequality and (4.4), we find

-k Ω u 2 ln |u|dx = -k Ω2 u 2 ln |u|dx -k Ω1 u 2 ln |u|dx ≤ -k Ω2 u 2 ln |u|dx = k Ω2 u 2 | ln |u||dx ≤ kd 0 Ω |u| 3 2 dx ≤ kd 0 |Ω| Ω |u| 3 dx 1 2
≤ kd 0 |Ω|c Since I(0) > 0 and I is continuous on [0, T ], there exists t 0 ∈ (0, T ] such that I(t) > 0, for all t ∈ [0, t 0 ). Let us denote by t 0 the largest real number in (0, T ] such that I > 0 on [0, t 0 ). If t 0 = T , then (4.6) is satisfied. We assume by contradiction that t 0 ∈ (0, T ). Thus I(t 0 ) = 0 and ∆u(t) Consequently, if g > 0 on [0, t 0 ), we get ∆u(s) 2 = 0, ∀s ∈ [0, t 0 ).

Then I(t) = 0, ∀t ∈ [0, t 0 ), which is not true since I > 0 on [0, t 0 ). If g = 0 on [0, t 0 ), then let t 1 ∈ [0, t 0 ) the smallest real number such that g(t 1 ) = 0. Because g(0) > 0 and g is positive, nonincreasing and continuous on R + (condition (A1)), then t 1 > 0 and g = 0 on [t 1 , ∞). Therefore, from (4.10), we deduce that 0 = t0 0 g(s) ∆u(s) 2 2 ds = t1 0 g(s) ∆u(s) 2 2 ds, then ∆u(s) 2 = 0, for any s ∈ [0, t 1 ), which implies that I(t) = 0, for any t ∈ [0, t 1 ). As in above, this is a contraduction with the fact that I > 0 on [0, t 0 ). Then we conclude that ∆u(t 0 ) 2 2 > 0. On the other hannd, we have

I(t 0 ) ≥ ∆u(t 0 ) 2 2 -3k Ω u(t 0 ) 2 ln |u(t 0 )|dx.
By using (4.9) and Lemma 4.1, we have

I(t 0 ) ≥ -3kc 3 * 6E(0) 1 2 ∆u(t 0 ) 2 2 .
By recalling (4.5), we arrive at I(t 0 ) > 0, which contradicts the assumption I(t 0 ) = 0. Hence, t 0 = T and then

I(t) > 0, ∀t ∈ [0, T ).

Stability

In this section, we state and prove our stability result. We start by establishing several lemmas needed for the proof of our main result. Lemma 5.1. Assume that g satisfies (A1). Then, for u ∈ H 2 0 (Ω), we have

Ω t 0 g(t -s)(u(t) -u(s))ds 2 dx ≤ c(go∆u)(t)
and

Ω t 0 g (t -s)(u(t) -u(s))ds 2 dx ≤ -c(g o∆u)(t).
Proof.

Ω t 0 g(t -s)(u(t) -u(s))ds 2 dx = Ω t 0 g(t -s) g(t -s)(u(t) -u(s))ds 2 dx.
By applying Cauchy-Schwarz' and Poincaré's inequalities, we can show that

Ω t 0 g(t -s)(u(t) -u(s))ds 2 dx ≤ Ω t 0 g(t -s)ds t 0 g(t -s)(u(t) -u(s)) 2 ds dx ≤ (1 -)c(go∆u)(t)
≤ c(go∆u)(t).

(5.1)

Similarly, the second inequality in Lemma 5.1 can be proved.

Lemma 5.2. Assume that g satisfies (A1) and (A2).

Then ∞ 0 ξ(t)g 1-σ (t)dt < ∞, ∀σ < 2 -p. (5.2)
Proof. Using (A1) and (A2), we easily see that, for any σ < 2 -p,

ξ(t)g 1-σ (t) = ξ(t)g 1-σ (t)g p (t)g -p (t) ≤ -g (t)g 1-σ-p (t).
Integrate the last inequality over (0, ∞), we obtain

∞ 0 ξ(t)g 1-σ (t)dt ≤ - ∞ 0 g (t)g 1-σ-p (t)dt = - g 2-p-σ (t) 2 -p -σ ∞ 0 < ∞.
Similar to Cavalcanti and Oquendo [START_REF] Cavalcanti | Frictional versus viscoelastic damping in a semilinear wave equation[END_REF], we can easily have the following lemma:

Lemma 5.3. Assume that (A1) -(A3) and (4.5) hold and u is a solution of (1.1).

Then, for any 0 < σ < 1, we have

(go∆u)(t) ≤ c t 0 g 1-σ (t)dt E(0) p-1 p-1+σ (g p o∆u) σ p-1+σ .
By taking σ = 1 2 , we get

(go∆u)(t) ≤ c t 0 g 1 2 (s)ds 2p-2 2p-1 (g p o∆u) 1 2p-1 (t) (5.3)
and, for any 0 ∈ (0, 1), (go∆u)

1 1+ 0 (t) ≤ c 1 1+ 0 t 0 g 1 2 (s)ds 2p-2 (2p-1)(1+ 0 ) (g p o∆u) 1 (2p-1)(1+ 0 ) (t).
(5.4)

Corollary 5.4. Assume that (A1) -(A3) and (4.5) hold and u is a solution of (1.1). Then

ξ(t)(go∆u)(t) ≤ c (-E (t)) 1 2p-1
(5.5) and, for any 0 ∈ (0, 1),

ξ(t)(go∆u) 1 1+ 0 (t) ≤ c 0 (-E (t)) 1 (2p-1)(1+ 0 ) .
(5.6)

Proof. Multiply both sides of (5.3) by ξ(t) and use (5.2) and (2.5) to obtain

ξ(t)(go∆u)(t) ≤ cξ 2p-2 2p-1 (t) t 0 g 1 2 (s)ds 2p-2 2p-1 ξ 1 2p-1 (t)(g p o∆u) 1 2p-1 (t) ≤ c t 0 ξ(s)g 1 2 (s)ds 2p-2 2p-1 (ξg p o∆u) 1 2p-1 (t) ≤ c ∞ 0 ξ(s)g 1 2 (s)ds 2p-2 2p-1 (-g o∆u) 1 2p-1 (t) ≤ c (-E (t)) 1 2p-1 .
(5.7)

For the proof of (5.6), using (5.5) and because ξ is nonnegative and nonincreasing, we obtain

ξ(t)(go∆u) 1 1+ 0 (t) = ξ 0 1+ 0 (t) (ξ(t)(go∆u)(t)) 1 1+ 0 ≤ c 0 (-E (t)) 1 (2p-1)(1+ 0 ) .
Lemma 5.5. Under the assumptions (A1) -(A3) and (4.5), the functional

ψ(t) := 1 ρ + 1 Ω |u t | ρ u t udx + Ω ∆u∆u t dx
satisfies, along the solution of (1.1), the estimate

ψ (t) ≤ - 2 Ω |∆u| 2 dx + Ω |∆u t | 2 dx + 1 ρ + 1 Ω |u t | ρ+2 dx + c(go∆u)(t) + k Ω u 2 ln |u|dx.
(5.8)

Proof. Direct differentiation of ψ, using (1.1), yields

ψ (t) = - Ω |∆u| 2 dx + Ω ∆u(t) t 0 g(t -s)∆u(s)dsdx + Ω |∆u t | 2 dx + 1 ρ + 1 Ω |u t | ρ+2 dx + k Ω u 2 ln |u|dx.
(5.9)

We then estimate the second term on the right side of (5.9). We have, using (2.1),

Ω ∆u(t) t 0 g(t -s)∆u(s)dsdx = Ω ∆u(t) t 0 g(t -s) (∆u(s) -∆u(t) + ∆u(t)) dsdx ≤ (1 -) Ω |∆u| 2 dx - Ω t 0 g(t -s) (∆u(t) -∆u(s)) ds dx.
By exploiting Lemma 5.1 and

ab ≤ 1 2η a 2 + η 2 b 2 , ∀a, b ≥ 0, ∀η > 0, we arrive at Ω ∆u(t) t 0 g(t -s)∆u(s)dsdx ≤ (1 -) Ω |∆u| 2 dx + 1 2η Ω t 0 g(t -s)|∆u(t) -∆u(s)|ds 2 dx + η 2 Ω |∆u| 2 dx ≤ 1 -+ η 2 Ω |∆u| 2 dx + c η (go∆u)(t).
By taking η = , we find

Ω ∆u(t) t 0 g(t -s)∆u(s)dsdx ≤ 2 - 2 Ω |∆u| 2 dx + c(go∆u)(t). (5.10)
Inserting (5.10) in (5.9), estimate (5.8) is established.

Lemma 5.6. Under the assumptions (A1) -(A3) and (4.5), the functional

χ(t) := - Ω ∆ 2 u t + 1 ρ + 1 |u t | ρ u t t 0 g(t -s)(u(t) -u(s))dsdx
satisfies, along the solution of (1.1) and for any δ, δ 1 , δ 2 > 0, the estimate

χ (t) ≤ (1 + 2(1 -) 2 )δ 1 + δ 4 Ω |∆u| 2 dx - 1 ρ + 1 t 0 g(s)ds Ω |u t | ρ+2 dx + c δ 1 + 1 δ 1 + 1 δ (go∆u)(t) - c δ 2 (g o∇u)(t) + δ 2 + cδ 2 (E(0)) ρ - t 0 g(s)ds Ω |∆u t | 2 dx + c 0,δ (go∆u) 1 1+ 0 (t).
(5.11)

Proof. Differentiating χ with respect to t and making use of (1.1), we find

χ (t) = Ω ∆u(t) t 0 g(t -s)(∆u(s) -∆u(t))dsdx - Ω t 0 g(t -s)∆u(s)ds t 0 g(t -s)(∆u(s) -∆u(t))ds dx - t 0 g(s)ds Ω |∆u t | 2 dx - Ω ∆u t (t) t 0 g (s)(∆u(s) -∆u(t))dsdx - 1 ρ + 1 Ω |u t | ρ u t t 0 g (t -s)(u(s) -u(t))dsdx - 1 ρ + 1 t 0 g(s)ds Ω |u t | ρ+2 dx -k Ω u ln |u| t 0 g(t -s) (u(t) -u(s)) dsdx.
(5.12) Now we proceed, using repeatedly Cauchy-Schwarz' inequality, Young's inequality and Lemma 5.1, to estimate each term in the right-hand side of (5.12). The first term may be estimated as follows

Ω ∆u(t) t 0 g(t -s)(∆u(s) -∆u(t))dsdx ≤ δ 1 Ω |∆u| 2 dx + c δ 1 (go∆u)(t), ∀δ 1 > 0.
(5.13)

For the second term, we recall (2.1) and the fact that (a + b) 2 ≤ 2(a 2 + b 2 ) to get, for any δ 1 > 0,

- Ω t 0 g(t -s)∆u(s)ds t 0 g(t -s)(∆u(s) -∆u(t))ds dx ≤ δ 1 Ω t 0 g(t -s)∆u(s)ds 2 dx + 1 4δ 1 Ω t 0 g(t -s)(∆u(s) -∆u(t))ds 2 dx ≤ δ 1 Ω t 0 g(t -s) ∆u(s) -∆u(t) +|∆u(t)| ds 2 dx + c δ 1 (go∆u)(t) ≤ c δ 1 + 1 δ 1 (go∆u)(t) + 2δ 1 (1 -) 2 Ω |∆u| 2 dx.
(5.14) For the fourth term, it is easy to see that, for any δ 2 > 0,

- Ω ∆u t t 0 g (t -s)(∆u(s) -∆u(t))dsdx ≤ δ 2 Ω |∆u t | 2 dx + c δ 2 Ω t 0 (-g (t -s)) |∆u(s) -∆u(t)| 2 dsdx.
(5.15)

The fifth term may be handled similarly

- 1 ρ + 1 Ω |u t | ρ u t t 0 g (t -s)(u(s) -u(t))dsdx ≤ 1 ρ + 1 δ 2 Ω |u t | 2(ρ+1) dx + c δ 2 Ω t 0 (-g (t -s)) |∆u(s) -∆u(t)| 2 dsdx .
(5.16) Using (2.4), (2.5), (4.1), (4.6) and (4.8), we have

E(0) ≥ E(t) = J(t) + 1 ρ + 2 u t ρ+2 ρ+2 ≥ J(t) ≥ 1 6 ∆u t 2 2 ,
which gives ∆u t 2 2 ≤ 6E(0).

(5.17)

By exploiting the Sobolev embedding

H 1 0 (Ω) → L 2(ρ+1) (Ω), (5.18)
and (5.17), we obtain

Ω |u t | 2(ρ+1) dx ≤ c(E(0)) ρ ∆u t 2 2 . (5.19)
Therefore (5.16) takes the form

- 1 ρ + 1 Ω |u t | ρ u t +∞ 0 g (s)(u(t -s) -u(t))dsdx ≤ cδ 2 (E(0)) ρ ∆u t 2 2 - c δ 2 (g o∆u)(t).
(5.20)

Applying (2.10) for s = |u|, using the embedding of H 2 0 (Ω) in L ∞ (Ω) and performing the same calulactions as before, we get, for any δ 3 > 0 and any ε 0 ∈ (0, 1), Then, for k small enough, there exist positive constants ε and N such that the functional

-k Ω u ln |u| t 0 g(t -s)(u(t) -u(s))dsdx ≤ k Ω u 2 + d 0 |u| 1-0 t 0 g(t -s)(u(t) -u(s))dsdx ≤ c Ω |u| 2 t 0 g(t -s)(u(t) -u(s))ds dx + δ 3 Ω u 2 dx + c 0 ,δ3 Ω t 0 g(t -s)(u(t) -u(s))ds 2 1+ 0 dx ≤ cδ 3 ||∆u|| 2 2 + c δ 3 Ω t 0 g(t -s)(u(t) -u(s))ds 2 dx + c 0 ,δ3 Ω t 0 g(t -s)(u(t) -u(s))ds
L = N E + εψ + χ satisfies L ∼ E (5.23)
and, for any t 0 > 0, there exists a positive constant m such that

L (t) ≤ -mE(t) + c(go∆u)(t) + c 0 (go∆u) 1 1+ 0 (t), ∀t ≥ t 0 . (5.24)
Proof. To prove (5.23), we use Young's inequality, the Sobolev embedding H 1 0 (Ω) → L ρ+2 (Ω), (4.6), (4.8), (5.17) and (5.19) to obtain

|L(t) -N E(t)| ≤ ε ρ + 2 ||u t || ρ+2 ρ+2 + ε (ρ + 1)(ρ + 2) ||u|| ρ+2 ρ+2 + ε 2 ||∆u t || 2 2 + ε 2 ||∆u|| 2 2 + 1 2(ρ + 1) ||u t || 2(ρ+1) 2(ρ+1) + 1 - 2(ρ + 1) c p (go∆u)(t) + 1 2 ||∆u t || 2 2 + 1 - 2 (go∆u)(t) ≤ εE(t) + ε c ρ+2 (ρ + 1)(ρ + 2) 6 E(0) 1+ ρ 2 E(t) + 3εE(t) + 3ε E(t) + 3c (ρ + 1) (E(0)) ρ E(t) + 3(1 -) (ρ + 1) c p E(t) + 3E(t) + 3(1 -)E(t) ≤ c(1 + ε)E(t); (5.25) that is [N -c(1 + ε)] E(t) ≤ L(t) ≤ [N + c(1 + ε)] E(t).
By fixing N large enough so that N > c(1 + ε), we obtain the desired result (5.23). For the proof of (5.24), since g is positive and g(0) > 0 then, for any t 0 > 0, we have t 0 g(s)ds ≥ t0 0 g(s)ds = g 0 > 0, ∀t ≥ t 0 .

By using (2.5), (5.8) and (5.11) then, for t ≥ t 0 , we have

L (t) ≤ N 2 - c δ 2 (g o∆u)(t) - g 0 -ε ρ + 1 Ω |u t | ρ+2 dx -ε 2 -(1 + 2(1 -) 2 )δ 1 - δ 4 ∆u 2 2 -[g 0 -ε -δ 2 -cδ 2 (E(0)) ρ ] ∆u t 2 2 + c ε + δ 1 + 1 δ 1 + 1 δ (go∆u)(t) + c 0,δ (go∆u) 1 1+ 0 (t) + εk Ω u 2 ln |u|dx.
(5.26)

Using the definition of E(t), we obtain, for any m > 0,

L (t) ≤ -mE(t) + N 2 - c δ 2 (g o∆u)(t) - g 0 -ε ρ + 1 - m ρ + 2 Ω |u t | ρ+2 dx -ε 2 -(1 + 2(1 -) 2 )δ 1 - δ 4 - m(1 -g 0 ) 2 ∆u 2 2 -g 0 -ε -δ 2 -cδ 2 (E(0)) ρ - m 2 ∆u t 2 2 + c ε + δ 1 + 1 δ 1 + 1 δ + m 2 (go∆u)(t) + c 0 ,δ (go∆u) 1 1+ 0 (t) + mk 4 u 2 2 + ε - m 2 k Ω u 2 ln |u|dx.
(5.27) Using the Logarithmic Sobolev inequality (2.7), we get, for 0 < m < 2ε,

L (t) ≤ -mE(t) + N 2 - c δ 2 (g o∆u)(t) - g 0 -ε ρ + 1 - m ρ + 2 Ω |u t | ρ+2 dx -ε 2 -(1 + 2(1 -) 2 )δ 1 - δ 4 - m(1 -g 0 ) 2 -ε - m 2 kc p a 2 2π ∆u 2 2 -g 0 -ε -δ 2 -cδ 2 (E(0)) ρ - m 2 ∆u t 2 2 + c ε + δ 1 + 1 δ 1 + 1 δ + m 2 (go∆u)(t) + c 0,δ (go∆u) 1 1+ 0 (t) -ε - m 2 k 2 2(1 + ln a) -ln u 2 2 u 2 2 + mk 4 u 2 2 .
(5.28) At this point, we choose our constant carefully. First, we pick 0 < ε < g 0 , then δ 1 , δ 2 and δ small enough so that

k 1 := ε 2 -(1 + 2(1 -) 2 )δ 1 - δ 4 > 0 and k 2 := g 0 -ε -δ 2 -cδ 2 (E(0)) ρ > 0.
Then, N sufficiently large so that

N > c(1 + ε) and N 2 - c δ 2 ≥ 0.
Consequently, we get

L (t) ≤ -mE(t) - g 0 -ε ρ + 1 - m ρ + 2 Ω |u t | ρ+2 dx -k 1 - m(1 -g 0 ) 2 -ε - m 2 kc p a 2 2π ∆u 2 2 -k 2 - m 2 ∆u t 2 2 + c + m 2 (go∆u)(t) + c 0 (go∆u) 1 1+ 0 (t) + mk 4 u 2 2 -ε - m 2 k 2 2(1 + ln a) -ln u 2 2 u 2 2 .
(5.29)

Finally, we choose m and k small enough so that m ≤ ε (so

mk 4 ≤ ε -m 2 k 2 ), g 0 -ε ρ + 1 - m ρ + 2 > 0, k 1 - m(1 -g 0 ) 2 -ε - m 2 kc p a 2 2π > 0 and k 2 - m 2 > 0,
we get

L (t) ≤ -mE(t) + c(go∆u)(t) + c ε0 (go∆u) 1 1+ε 0 (t) -ε - m 2 k 2 1 + 2 ln a -ln u 2 2 u 2 2 .
( So, from (2.5) and using Young's inequality, we get

|E (t)| = 1 2 g(t) ∆u(t) 2 2 - 1 2 (g o∆u)(t) ≤ 1 2 g(t) ∆u(t) 2 2 - t 0 g (t -s) ∆u(t) 2 2 + ∆u(s) 2 2 ds ≤ 6 l 1 2 g(t) + 2g(0) -2g(t) E(0)
≤ cE(0).

(5.33)

Theorem 5.9. Let (u 0 , u 1 ) ∈ H 2 0 (Ω) × H 2 0 (Ω), ∈ (0, 2p -1) and t 0 > 0. Assume that (A1) -(A3) and (4.5) hold. Then, for k small enough, there exists a positive constant K such that the solution of (1.1) satisfies

E(t) ≤ K 1 + t t0 ξ 2p-1+ (s)ds -1 2p-2+
, ∀t ≥ t 0 .

(5.34)

Moreover, if there exist 1 ∈ (0, 2p -1) and t 0 > 0 such that

∞ t0 1 + t t0 ξ 2p-1+ 1 (s)ds -1 2p-2+ 1 dt < ∞, (5.35)
then, for any r ∈ (0, p) and t 0 > 0, there exists a positive constant K such that the solution of (1.1) satisfies

E(t) ≤ K 1 + t t0 ξ p+r (s)ds -1 p-1+r
, ∀t ≥ t 0 .

(

ξ(t)L (t) ≤ -mξ(t)E(t) + c (-E (t)) 1 2p-1 + c (-E (t)) 1 (2p-1)(1+ 0 ) ≤ -mξ(t)E(t) + c (-E (t)) 0 (2p-1)(1+ 0 ) (-E (t)) 1 (2p-1)(1+ 0 ) + c (-E (t)) 1 (2p-1)(1+ 0 ) ≤ -mξ(t)E(t) + c (-E (t)) 1 (2p-1)(1+ 0 ) , ∀t ≥ t 0 .
(5.38) Multiply the last inequality by ξ γ (t)E γ (t), where γ = (2p -1)(1 + 0 ) -1, and notice that ξ ≤ 0 to obtain

ξ γ+1 (t)E γ (t)L (t) ≤ -mξ γ+1 (t)E γ+1 (t) + c (ξE) γ (t) (-E (t)) 1 γ+1 , ∀t ≥ t 0 .
Use of Young's inequality, with q = γ + 1 and q * = γ+1 γ , gives, for any ε > 0,

ξ γ+1 (t)E γ (t)L (t) ≤ -mξ γ+1 (t)E γ+1 (t) + c ε ξ γ+1 (t)E γ+1 -c ε E (t) = -(m -ε c)ξ γ+1 (t)E γ+1 -cE (t), ∀t ≥ t 0 .
We then choose 0 < ε < m c and recall that ξ ≤ 0 and E ≤ 0, to get, for c 1 = m -ε c, ξ γ+1 E γ L (t) ≤ ξ γ+1 (t)E γ (t)L (t) ≤ -c 1 ξ γ+1 (t)E γ+1 (t) -cE (t), ∀t ≥ t 0 , which implies ξ γ+1 E γ L + cE (t) ≤ -c 1 ξ γ+1 (t)E γ+1 (t), ∀t ≥ t 0 .

Let F = ξ γ+1 E γ L + cE. Then F ∼ E (thanks to (5.23)) and F (t) ≤ -cξ γ+1 (t)F γ+1 (t) = -cξ (2p-1)(1+ 0 ) (t)F (2p-1)(1+ 0 ) (t), ∀t ≥ t 0 .

Integrating over (t 0 , t) and using the fact that F ∼ E, we obtain (5.34) with = (2p -1) 0 . To establish (5.36), we use the idea of Messaoudi and Al-Khulaifi [START_REF] Messaoudi | General and optimal decay for a quasilinear viscoelastic equation[END_REF] (5.42) Inequality (5.38) with 2p -1 replaced by p is exactely (5.42). Then, the proof of (5.36) can be completed as for the one of (5.34) (by taking γ = p(1 + 0 ) -1 and = p 0 ). This completes the proof of our main result.

Remarks 5.11. We note here that 2p -2 + and p -1 + can be arbitrary close to 2p -2 and p -1, respectively, since can be arbitrary close to zero. On the other hand, in the absence of the logarithmic "forcing" term (k = 0), the estimates (2.10) and (5.21) drop out and, consequently, (5.24) takes the form L (t) ≤ -mE(t) + c(go∆u)(t), ∀t ≥ t 0 .

(5.43)

In this case, we obtain the following result:

Theorem 5.12. Let (u 0 , u 1 ) ∈ H 2 0 (Ω) × H 2 0 (Ω) and t 0 > 0. Assume that (A1) -(A2) hold. Then, there exists a positive constant K such that the solution of (1. , ∀t ≥ t 0 .

(5.47)

2 2

 2 dt ≤ C 4 . (3.16) From (3.11) and (3.16), we have

  taking into account the Lebesgue bounded convergence theorem (Ω is bounded), we get ku m ln |u m | → ku ln |u| strongly in L 2 (0, T ; L 2 (Ω)).

  ln |u m (s)|dxds. (3.24) Convergences (3.2), (3.18), (3.20) and (3.23) are sufficient to pass to the limit in (3.24) as m → ∞, and get, for any w ∈ V m and m ≥ 1,

  ) which implies that(3.25) is valid for any w ∈ H 2 0 (Ω). Using the fact that the terms in the right-hand side of (3.25) are absolutely continuous (since they are functions of t defined by integrals over (0, t)), then (3.25) is differentiable for a.e. t ∈ R + . Thus, differentiating (3.25), we obtain, for a.e. t ∈ (0, T ) and w ∈ H 2 0 (Ω), Ω |u t | ρ u tt wdxds + -s)∆u(s) ∆wdsdx = k Ω wu(t) ln |u(t)|dxds. (3.26)

δ 4 =Ω u ln |u| t 0 gLemma 5 . 7 .

 4057 cδ 3 and using Holder's inequality and Lemma 5.1, we find -k (t -s)(u(t) -u(s))dsdx ≤ 12)-(5.15), (5.20) and (5.21), estimate (5.11) is established. Assume that (A1) -(A3) and (4.5) hold and let ε 0 ∈ (0, 1).

≤ 1 p

 1 (ξ 0 (0)ξ(t)(g • ∆u)(t))1 1+ 0 ≤ c (ξ(t)(g • ∆u)(t)) 1 1+ 0 ≤ c(-E (t)) 1 p(1+ 0 ) . (5.41) If η(t) = 0, then s → ∆u(s) is a constant function on [0, t]. Therefore (g • ∆u)(t) = 0,and hence (5.40) and (5.41) hold. Now, multiplying (5.24) by ξ(t) and using (5.33), (5.40) and (5.41) to find, for any t ≥ t 0 (as for (5.38)),ξ(t)L (t) ≤ -mξ(t)E(t) + c (-E (t)) + c (-E (t)) 1 p(1+ 0 ) ≤ -mξ(t)E(t) + c (-E (t)) 0 p(1+ 0 ) (-E (t)) 1 p(1+ 0 ) + c (-E (t)) 1 p(1+ 0 )≤ -mξ(t)E(t) + c (-E (t)) 1 p(1+ 0 ) , ∀t ≥ t 0 .

  [START_REF] Barrow | Inflationary models with logarithmic potentials[END_REF] satisfies, for all t ≥ t 0 ,

	E(t) ≤ Ke	-λ t t 0	ξ(s)ds	if p = 1		(5.44)
	and
	E(t) ≤ K 1 +	t t0	ξ 2p-1 (s)ds	-1 2p-2	if 1 < p <	3 2	.	(5.45)
	Moreover, if 1 < p < 3 2 and
	∞		t		-1 2p-2
	1 +		ξ 2p-1 (s)ds	dt < ∞,		(5.46)
	0		t0
	then
				t		-1 p-1
	E(t) ≤ K 1 +	ξ p (s)ds
				t0

Acknowledgment

The authors thank KFUPM for its continuous support. This work was funded by KFUPM under Project #IN151023. This work was initiated during the visit of the second author to KFUPM in December 2016.