Mohammad M Al-Gharabli

King Fahd

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

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

EXISTENCE AND A GENERAL DECAY RESULTS FOR A VISCOELASTIC PLATE EQUATION WITH A LOGARITHMIC NONLINEARITY

Keywords: 2000 Mathematics Subject Classification. 35B35, 35L55, 75D05, 74D10, 93D20 General decay, logarithmic nonlinearity, viscoelasticity, plate equation

 Hiramatsu et al. [27] and Han and Wang [26].

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

     u tt + ∆ 2 u + u - t 0 g(t -s)∆ 2 u(s)ds = ku ln |u|,
in Ω × (0, ∞), u = ∂u ∂ν = 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 k is a small postive real number. The kernel g satisfies some conditions to be specified later.

1.1. 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] 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, (1.2)
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] Chen | 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 system:

               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, (1.3) 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]. 1.2. Viscoelastic problems. Since the pioneer works of Dafermos [START_REF] Dafermos | Asymptotic stability in viscoelasticity[END_REF][START_REF] Dafermos | On abstract volterra equations with applications to linear viscoelasticity[END_REF] in 1970, where the general decay was discussed, problems related to viscoelasticity have attracted a great deal of attention and many results of existence and long-time behavior have been established. The importance of the viscoelastic properties of materials has been realized because of the rapid developments in rubber and plastic industry. Many advances in the studies of constitutive relations, failure theories and life prediction of viscoelastic materials and structures were reported and reviewed in the last two decades [START_REF] Christensen | Theory of Viscoelasticity, An Introduction[END_REF]. Hrusa [START_REF] Hrusa | Global existence and asymptotic stability for a semilinear Volterra equation with large initial data[END_REF] considered a one-dimensional nonlinear viscoelastic equation of the form

u tt -cu xx + t 0 m(t -s)(ψ(u x (s))) x ds = f (x, t)
and proved several global existence results for large data and an exponential decay result for strong solutions when m(s) = e -s and ψ satisfies certain conditions. In [START_REF] Dasios | Equipartition of energy in linearized 3-D viscoelasticity[END_REF], Dassios and Zafiropoulos considered a viscoelastic problem in R 3 and proved a polynomial decay result for exponentially decaying kernels. After that, Rivera [START_REF] Rivera | Asymptotic behavior in linear viscoelasticity[END_REF] considered equations for linear isotropic homogeneous viscoelastic solids of integral type which occupy bounded domains or the whole space R n . In the boundeddomain case and for exponentially decaying memory kernels and regular solutions, he showed that the sum of the first and the second energy decays exponentially. Whereas, the decay is polynomial when the body occupies the whole space R n , even if the relaxation function is of an exponential decay.

For quasilinear problems, Cavalcanti et al. [START_REF] Cavalcanti | Existence and uniform decay for nonlinear viscoelastic equation with strong damping[END_REF] studied, in a bounded domain, the following equation:

|u t | ρ u tt -∆u -∆u tt + t 0 g(t -τ )∆u(τ )dτ -γ∆u t = 0, in Ω × (0, ∞), (1.4)
for ρ > 0. A global existence result for γ ≥ 0, as well as an exponential decay result for γ > 0, have been established. This latter result was then extended to a situation, where γ = 0, by Messaoudi and Tatar [START_REF] Messaoudi | Global existence asymptotic behavior for a non-linear viscoelastic problem[END_REF][START_REF] Messaoudi | Global existence and uniform stability of solutions for a quasilinear viscoelastic problem[END_REF], and exponential and polynomial decay results have been established in the absence, as well as in the presence, of a source term. In all the above mentioned works, the rates of decay in relaxation functions were either of exponential or polynomial type. In [START_REF] Cavalcanti | Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping[END_REF], Cavalcanti et al. considered

u tt -∆u + t 0 g(t -s)∆u(s)ds + a(x)u t + |u| p-1 u = 0, in Ω × (0, ∞),
where a : Ω → R + is a function which may vanish on a part of the domain Ω but satisfies a(x) ≥ a 0 on ω ⊂ Ω and g satisfies, for two positive constants ξ 1 and ξ 2 ,

-ξ 1 g(t) ≤ g (t) ≤ -ξ 2 g(t), ∀t ∈ R +
and established an exponential decay result under some restrictions on ω. Berrimi and Messaoudi [START_REF] Berrimi | Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping[END_REF] established the result of [START_REF] Cavalcanti | Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping[END_REF], under weaker conditions on both a and g, to a problem where a source term is competing with the damping term. Cavalcanti and Oquendo [START_REF] Cavalcanti | Frictional versus viscoelastic damping in a semilinear wave equation[END_REF] considered the following problem:

u tt -k 0 ∆u+ t 0 div[a(x)g(t -s)∆u(s)]ds+b(x)h(u t )+f (u) = 0, in Ω×(0, ∞) (1.5)
and established, for a(x) + b(x) ≥ ρ > 0, an exponential stability result for g decaying exponentially and h linear, and a polynomial stability result for g decaying polynomially and h nonlinear. Li et al. [START_REF] Li | Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation[END_REF] treated (1.5) with b(x) = 0 and f (u) = -|u| γ u, γ > 0. They showed the global existence and uniqueness of global solution of problem (1.5) and established uniform decay rate of the energy under suitable conditions on the initial data and the relaxation function g. For more general decaying relaxation functions, 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] considered

u tt -∆u + t 0 g(t -τ )∆u(τ )dτ = b|u| q-2 u, in Ω × (0, ∞) (1.6)
for q ≥ 2, b ∈ {0, 1} and g satisfying (A1) and (A2) below with p = 1, and established a more general decay result, from which the usual exponential and polynomial decay rates are only special cases. Guesmia et al. [START_REF] Guesmia | General decay of solutions of a nonlinear system of viscoelastic wave equations[END_REF] studied the well-posedness and stability of the following coupled two wave equations:

u tt -∆u + t 0 g 1 (t -τ )∆u(τ )dτ + |u t | m-1 u t = f 1 (u, v), in Ω × (0, ∞), v tt -∆v + t 0 g 2 (t -τ )∆v(τ )dτ + |v t | r-1 v t = f 2 (u, v), in Ω × (0, ∞), (1.7)
where m, r ≥ 1, f 1 and f 2 are given functions satisfying some hypotheses, and g 1 and g 2 are like g in (A1) and (A2) below (with p = 1, ξ 1 and ξ 2 instead of ξ). They established the same stability estimate as in [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] with ξ = min{ξ 1 , ξ 2 }. Very recently, Messaoudi and Al-Khulaifi [START_REF] Messaoudi | General and optimal decay for a quasilinear viscoelastic equation[END_REF] considered (1.4) with γ = 0, where the relaxation function satisfies (2.2) below, and established a more general decay result. For the case of memories acting on the boundary of domain, we refer the readers to [START_REF] Cavalcanti | General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type[END_REF][START_REF] Guesmia | Stabilisation de l'équation des ondes avec conditions aux limites de type mémoire[END_REF] and the references therein.

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], (1.8)
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 (1.9)
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 × L 2 to the initial-boundary value problem (1.9) 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 onedimensional Cauchy problem for equation (1.9). 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| (1.10)
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 (1.10) in Ω ⊂ R 3 .

In this paper, we are concerned with the well-posedness and stability of the plate problem (1.1) with a kernel 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 notations and material needed in 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 and 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) satisfies 0 < k < k 0 , where k 0 is the positive real number satisfying 2π k 0 c p = e -3 2 -1 k 0 (2.3)
and 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) . Remark 2.1. The function f (s) = 2π cps -e -3 2 -1
s is a continuous and decreasing function on (0, ∞), with

lim s→0 + f (s) = ∞ and lim x→∞ f (x) = -e -3 2 .
Then, there exists a unique

k 0 > 0 such that f (k 0 ) = 0. Moreover, e -3 2 -1 s < 2π c p s , ∀s ∈ (0, k 0 ). (2.4)
The energy functional associated with problem (1.1) is

E(t) = 1 2 u t 2 2 + 1 - t 0 g(s)ds ∆u 2 2 + k + 2 2 u 2 2 - 1 2 Ω u 2 ln |u| k dx + 1 2 (go∆u)(t), (2.5)
where

(go∆u)(t) = t 0 g(t -s) ∆u(s) -∆u(t) 2 2 ds.
Direct differentiation of (2.5), using (1.1), leads to [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF][START_REF] Chen | Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity[END_REF], Logarithmic Sobolev inequality). Let u be any function in H 1 0 (Ω) and a be any positive real number. Then

E (t) = 1 2 (g o∆u)(t) - 1 2 g(t) ∆u 2 2 ≤ 1 2 (g o∆u)(t) ≤ 0. (2.6) Lemma 2.2 ([
Ω u 2 ln |u|dx ≤ 1 2 u 2 2 ln u 2 2 + a 2 2π ∇u 2 2 -(1 + ln a) u 2 2 .
(2.7)

Corollary 2.3. Let u be any function in H 2 0 (Ω) and a be any positive real number. Then

Ω u 2 ln |u|dx ≤ 1 2 u 2 2 ln u 2 2 + c p a 2 2π ∆u 2 2 -(1 + ln a) u 2 2 .
(2.8)

Lemma 2.4 ([11], Logarithmic Gronwall inequality). Let c > 0, γ ∈ L 1 (0, T ; R + )
and assume that the function w

: [0, T ] → [1, ∞) satisfies w(t) ≤ c 1 + t 0 γ(s)w(s) ln w(s)ds , 0 ≤ t ≤ T. (2.9) Then w(t) ≤ c exp c t 0 γ(s)ds , 0 ≤ t ≤ T.
(2.10)

3. 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([0, T ], H 2 0 (Ω)) ∩ C 1 ([0, T ], L 2 (Ω)) ∩ C 2 ([0, T ], H -2 (Ω)) is called a weak solution of (1.1) on [0, T ] if, for any w ∈ H 2 0 (Ω) and t ∈ [0, T ],      Ω u tt (x, t)w(x)dx + Ω ∆u(x, t)∆w(x)dx + Ω u(x, t)w(x)dx -Ω ∆w(x) t 0 g(t -s)∆u(s)dsdx = Ω u(x, t)w(x) ln |u(x, t)| k dx, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x). (3.1)
Theorem 3.2. Assume that (A1) and (A3) hold and let

(u 0 , u 1 ) ∈ H 2 0 (Ω) × L 2 (Ω). Then problem (1.1) has a weak solution u ∈ C([0, T ], H 2 0 (Ω)) ∩ C 1 ([0, T ], L 2 (Ω)) ∩ C 2 ([0, T ], H -2 (Ω)). (3.2)
Proof. To establish the existence of a solution to problem (1.1), we use the Faedo-Galerkin method. Let {w j } ∞ j=1 be an orthogonal basis of the "separable" space

H 2 0 (Ω) which is orthonormal in L 2 (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) and 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 L 2 (Ω), as m → ∞. (3.3)
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 tt w + ∆u m ∆w + u m w - t 0 g(t -s)∆u m (s)ds∆w dx = Ω wu m ln |u m | k 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 . (3.4)
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 (3.4) 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 independent of m and t. For this purpose, let us replace w by u m t in (3.4) and integrate by parts to obtain

d dt E m (t) ≤ 1 2 (g o∆u m )(t) ≤ 0, (3.5)
where

E m (t) = 1 2 u m t 2 2 + 1 - t 0 g(s) ∆u m 2 2 + k + 2 2 u m 2 2 - 1 2 Ω |u m | 2 ln |u m | k dx + 1 2 (go∆u m )(t).
(3.6) From (3.5), we have E m (t) ≤ E m (0). The last inequality together with (2.1) and the Logarithmic Sobolev inequality (2.8) lead to

u m t 2 2 + (go∆u m )(t) + - ka 2 c p 2π ∆u m 2 2 + k + 2 2 + k (1 + ln a) u m 2 2 ≤ C + k 2 u m 2 2 ln u m 2 2 , (3.7) where C = 2E m (0). Choosing e -3 2 -1 k < a < 2π kc p (3.8)
will make

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

(go∆u m )(t) + u m t 2 2 + ∆u m 2 2 + u m 2 2 ≤ c 1 + u m 2 2 ln u m 2 2 .
(3.9)

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, (3.10)
hence, inequality (3.9) gives

u m 2 2 ≤ 2 u m (0) 2 2 + 2cT 1 + t 0 u m 2 2 ln u m 2 2 ds . (3.11) If we put C 1 = max {2cT, 2 u m (0) 2 2 }, (3.11) leads to u m 2 2 ≤ 2C 1 1 + t 0 u m 2 2 ln u m 2 2 ds .
Without loss of generality, we take C 1 ≥ 1, which gives

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 (3.9) it follows that:

(go∆u m )(t) + u m t 2 2 + ∆u m 2 2 + u m 2 2 ≤ c (1 + C 2 ln C 2 ) := C 3
, where C 3 is a positive constant independent of m and t. This implies sup t∈(0,tm)

u m t 2 2 + sup t∈(0,tm) ∆u m 2 2 + sup t∈(0,tm) u m 2 2 ≤ 3C 3 . (3.12)
So, the approximate solution is uniformly bounded independent of m and t. Therefore, we can extend t m to T . Moreover, we obtain, from (3.12),

u m is uniformly bounded in L ∞ (0, T ; H 2 0 (Ω)), u m t is uniformly bounded in L ∞ (0, T ; L 2 (Ω)), (3.13)
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 2 (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 2 (Ω)). (3.14)
Making use of Aubin-Lions' theorem, we find, up to a subsequence, that

u m → u strongly in L 2 (0, T ; L 2 (Ω))
and u m → u a.e. in Ω × (0, T ).

Since the map s → s ln |s| k is continuous on R, we have the convergence

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

H 2 0 (Ω) in L ∞ (Ω) (since Ω ⊂ R 2 ), it is clear that u m ln |u m | k is bounded in L ∞ (Ω × (0, T )).
Next, taking into account the Lebesgue bounded convergence theorem (Ω is bounded), we get

u m ln |u m | k → u ln |u| k strongly in L 2 (0, T ; L 2 (Ω)). (3.15)
Now, we integrate (3.4) over (0, t) to obtain, for every

w ∈ V m , Ω u m t wdx - Ω u m 1 wdx + t 0 Ω ∆u m (s)∆wdxds + t 0 Ω u m (s)wdxds - t 0 Ω τ 0 g(τ -s)∆u m (s)ds∆wdxdsdτ = Ω t 0 wu m (s) ln |u m (s)| k dxds.
(3.16) Convergences (3.3), (3.14) and (3.15) are sufficient to pass to the limit in (3.16), as m → +∞, and get, for any w ∈ V m and m ≥ 1,

Ω u t wdx = Ω u 1 wdx - t 0 Ω ∆u(s)∆wdxds - t 0 Ω
u(s)wdxds

+ τ 0 Ω ∆w(x) t 0 g(t -s)∆u(s)dsdxdt + Ω t 0 u(s)w ln |u(s)| k dsdx, (3
.17) which implies that (3.17) is valid for any w ∈ H 2 0 (Ω). Using the fact that the terms in the right-hand side of (3.17) are absolutely continuous since they are functions of t defined by integrals over (0, t), hence it is differentiable for a.e. t ∈ R + . Thus, differentiating (3.17), we obtain, for a.e. t ∈ (0, T ) and any w ∈ H 2 0 (Ω),

Ω u tt (x, t)w(x)dx + Ω ∆u(x, t)∆w(x)dx + Ω u(x, t)w(x)dx - Ω ∆w(x) t 0 g(t -s)∆u(s)dsdx = Ω w(x)u(x, t) ln |u(x, t)| k dx. (3.18)
To handle the initial conditions, we note that u m u weakly in L 2 (0, T ; H 2 0 (Ω)) and u m t u t weakly in L 2 (0, T ; L 2 (Ω)).

(3.19) Thus, using Lion's Lemma [START_REF] Lions | Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires[END_REF], we obtain

u m → u in C([0, T ], L 2 (Ω)). (3.20)
Therefore, u m (x, 0) makes sense and

u m (x, 0) → u(x, 0) in L 2 (Ω).
Also, we have As m → ∞, we have, for any w ∈ H 2 0 (Ω) and any φ ∈ C ∞ 0 ((0, T )),

u m (x, 0) = u m 0 (x) → u 0 (x) in H 2 0 (Ω). Hence u(x, 0) = u 0 (x). Now,
- T 0 Ω u t (t)wφ (t)dxdt = - T 0 Ω ∆u(t)∆wφ(t)dxdt - T 0 Ω uwφ(t)dxdt + T 0 Ω t 0 g(t -s)∆u(s)ds∆wφ(t)dxdt + T 0 Ω wφ(t)u ln |u| k dxdt.
(3.22) This means (see [START_REF] Lacroix-Sonrier | Distrubutions Espace de Sobolev Application[END_REF]),

u tt ∈ L 2 ([0, T ), H -2 (Ω)). Recalling that u t ∈ L 2 ((0, T ), L 2 (Ω)), we obtain u t ∈ C([0, T ), H -2 (Ω)).
So, u m t (x, 0) makes sense and

u m t (x, 0) → u t (x, 0) in H -2 (Ω). But u m t (x, 0) = u m 1 (x) → u 1 (x) in L 2 (Ω). Hence u t (x, 0) = u 1 (x).
4. 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 2 2 + (go∆u)(t) - Ω u 2 ln |u| k dx + k 4 u 2 2 (4.1)
and

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

-kd 0 |Ω|c 3 * ∆u 3 2 2 ≤ Ω u 2 ln |u| k dx ≤ kc 3 * ∆u 3 2 , ∀u ∈ H 2 0 (Ω), (4.3)
where d 0 = sup 0<s<1 √ s| ln s|, |Ω| is the Lebegue 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)). Proof. Let Ω 1 = {x ∈ Ω : |u(x)| > 1} and Ω 2 = {x ∈ Ω : |u(x)| ≤ 1}.
So, using (4.4), we have

Ω u 2 ln |u| k dx = Ω2 u 2 ln |u| k dx + Ω1 u 2 ln |u| k dx ≤ k Ω1 u 2 ln |u|dx ≤ k Ω1 |u| 3 dx ≤ k Ω |u| 3 dx ≤ kc 3 * ∆u 3 2 ,
this gives the right inequality in (4.3). On the other hand, using Hölder's inequality and (4.4), we find

- Ω u 2 ln |u| k dx = - Ω2 u 2 ln |u| k dx - Ω1 u 2 ln |u| k 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 3 * ∆u 3 2 2 ,
which implies the left inequality in (4.3). Proof. From (4.2), we have

Lemma 4.2. Assume that (A1) -(A3). Let (u 0 , u 1 ) ∈ H 2 0 (Ω) × L 2 (Ω) such that I(0) > 0 and √ 54kc 3 * E(0)
Ω u 2 ln |u| k dx = 1 3 1 - t 0 g(s)ds ∆u 2 2 + 1 3 u 2 2 + 1 3 (go∆u)(t) - 1 3 I(t). (4.7)
Substitute (4.7) in (4.1), we find

J(t) = 1 3 1 - t 0 g(s)ds ∆u 2 2 + u 2 2 + (go∆u)(t) + k 4 u 2 2 + 1 6 I(t). (4.8)
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 biggest 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) 2 2 ≤ 3 J(t) ≤ 3 E(t) ≤ 3 E(0), ∀t ∈ [0, t 0 ). (4.9)
If ∆u(t 0 ) 2 2 = 0, then (4.3) and (4.4) give

0 = I(t 0 ) = (go∆u) (t 0 ) = t0 0 g(s) ∆u(s) 2 2 ds. (4.10)
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 is not positive 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 before, 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 hand, we have

I(t 0 ) ≥ ∆u(t 0 ) 2 2 -3 Ω u(t 0 ) 2 ln |u(t 0 )| k 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 ).
The global existence can be easily established by repeating the steps of the proof of Theorem 3.1 [START_REF] Messaoudi | Global existence and uniform stability of solutions for a quasilinear viscoelastic problem[END_REF].

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).

(

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 ∞ 0 g 1-σ (t)dt E(0) p-1 p-1+σ (g p o∆u) σ p-1+σ (t).
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 (2.6) 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 .
(

For the proof of (5.6), using (5.5) and because ξ is 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. Assume that (A1) -(A3) and (4.5) hold. Then the functional

ψ(t) = Ω uu t dx
satisfies, along the solutions of (1.1),

ψ (t) ≤ ||u t || 2 2 - 2 ||∆u|| 2 2 -||u|| 2 2 + Ω u 2 ln |u| k dx + c(go∆u)(t). (5.8)
Proof. By using Eq. (1.1), we easily see that

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

We now use Lemma 5.1 and Young's inequality, to obtain, for any µ > 0,

Ω ∆u(t) t 0 g(t -s)∆u(s)ds dx ≤ 1 -+ µ 2 ||∆u|| 2 2 + 1 2µ (1 -)(go∆u)(t).
(5.10)

By choosing µ = and combining (5.9) and (5.10), we obtain (5.8).

Lemma 5.6. Assume that (A1) -(A3) and (4.5) hold. Then the functional

χ(t) = - Ω u t t 0 g(t -s)(u(t) -u(s))dsdx
satisfies, along the solutions of (1.1) and for any 0 ∈ (0, 1) and δ > 0,

χ (t) ≤ δ||∆u|| 2 2 + c δ (go∆u)(t) + c δ (-g o∆u)(t) + δ - t 0 g(s)ds u t 2 2
+ c 0,δ (go∆u)

1 1+ 0 (t).
(5.11)

Proof. Direct computations, using (1.1), yield

χ (t) = Ω ∆u t 0 g(t-s)(∆u(t)-∆u(s))dsdx+ Ω u t 0 g(t-s)(u(t)-u(s))dsdx + Ω t 0 g(t -s)(∆u(t) -∆u(s))ds t 0 g(t -s)∆u(s)dsdx - Ω u ln |u| k t 0 g(t -s)(u(t) -u(s))dsdx - Ω u t t 0 g (t -s)(u(t) -u(s))dsdx - t 0 g(s)ds Ω u 2 t dx.
(5.12) Similarly to (5.9), we estimate the right-hand side terms of (5.12). So, by using Young's inequality, the first term gives, for any δ > 0, (5.16)

Ω
Let 0 ∈ (0, 1) and f (s) = s 0 (| ln s| -s). Notice that f is continuous on (0, ∞) and its limit at 0 is 0, and its limit at ∞ is -∞. Then f has a maximum d 0 on [0, ∞), so the following inequality holds:

s| ln s| ≤ s 2 + d 0 s 1-0 , ∀s > 0.

(5.17)

EXISTENCE AND A GENERAL DECAY FOR A VISCOELASTIC PLATE EQUATION 173

Applying this inequality to u ln |u|, using the embedding of H 2 0 (Ω) in L ∞ (Ω) and performing the same calulactions as before, we get, for any δ 1 > 0,

Ω u ln |u| k 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| t 0 g(t -s)(u(t) -u(s))ds dx + δ 1 Ω u 2 dx + c 0,δ1 Ω t 0 g(t -s)(u(t) -u(s))ds 2 1+ 0 dx ≤cδ 1 ||∆u|| 2 2 + c δ 1 Ω t 0 g(t -s)(u(t) -u(s))ds 2 dx + c 0 ,δ1 Ω t 0 g(t -s)(u(t) -u(s))ds 2 1+ 0 dx,
then, puting δ 4 = cδ 1 and using Hölder's inequality and Lemma 5.1, we find

Ω u ln |u| k t 0 g(t -s)(u(t) -u(s))dsdx ≤ δ 4 ||∆u|| 2 2 + c δ (go∆u)(t) + c 0,δ (go∆u) 1 1+ 0 (t).
(

The above inequalities imply (5.11).

Lemma 5.7. Assume that (A1) -(A3) and (4.5) hold and let 0 ∈ (0, 1). Then, for k small enough, there exist two positive constants ε 1 and ε 2 such that the functional

L(t) = E(t) + ε 1 ψ(t) + ε 2 χ(t) satisfies L ∼ E (5.19)
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 . (5.20)
Proof. For the proof of (5. [START_REF] Gorka | Logarithmic Klein-Gordon equation[END_REF]), we see that, using similar calculations as before,

|L(t) -E(t)| = |ε 1 ψ(t) + ε 2 χ(t)| ≤ c (ε 1 + ε 2 ) ||u t || 2 2 + ||∆u|| 2 2 + (go∆u)(t) ,
therefore, from (4.6) and (4.8), we obtain

|L(t) -E(t)| ≤ c (ε 1 + ε 2 ) 1 2 ||u t || 2 2 + J(t) = c (ε 1 + ε 2 ) E(t), then (1 -c (ε 1 + ε 2 )) E(t) ≤ L(t) ≤ (1 + c (ε 1 + ε 2 )) E(t).
Hence, for ε 1 , ε 2 > 0 satisfying

1 -c (ε 1 + ε 2 ) > 0, (5.21)
the equivalence (5.19) holds. Now, we prove inequality (5.20). 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 (2.6), (5.8), (5.11) and the definition of E(t), then, for t ≥ t 0 and any m > 0, we have

L (t) ≤ -mE(t) -ε 2 (g 0 -δ) -ε 1 - m 2 u t 2 2 - 2 ε 1 -ε 2 δ - m 2 ∆u 2 2 -ε 1 - (k + 2)m 4 u 2 2 + kε 1 -k m 2 Ω u 2 ln |u|dx + cε 1 + ε 2 c δ + m 2 (go∆u)(t) + 1 2 - cε 2 δ (g o∆u)(t) + ε 2 c 0,δ (go∆u) 1
1+ 0 (t).

(5.22)

Using the Logarithmic Sobolev inequality, for 0 < m < 2ε 1 , we get

L (t) ≤ -mE(t) -ε 2 (g 0 -δ) -ε 1 - m 2 u t 2 2 - 2 ε 1 -ε 2 δ - m 2 -k ε 1 - m 2 c p a 2 2π ∆u 2 2 -ε 1 - m(k + 2) 4 + k ε 1 - m 2 (1 + ln a) + k m 4 - ε 1 2 ln u 2 2 u 2 2 + cε 1 + ε 2 c δ + m 2 (go∆u)(t) + 1 2 - cε 2 δ (g o∆u)(t) + ε 2 c 0,δ (go∆u) 1 1+ 0 (t).
(5.23) At this point we choose δ so small that g 0 -δ > 1 2 g 0 and δ < g 0 16 .

Whence δ is fixed, the choice of any two positive constants ε 1 and ε 2 satisfying

g 0 4 ε 2 < ε 1 < g 0 2 ε 2 (5.24)
will make

k 1 := ε 2 (g 0 -δ) -ε 1 > 0 and k 2 := 2 ε 1 -ε 2 δ > 0.
Then, we choose ε 1 and ε 2 so small so that (5.21) and (5.24) remain valid and, further,

1 2 - cε 2 δ > 0.
Consequently, we get (5.19) and

L (t) ≤ -mE(t) -k 1 - m 2 u t 2 2 -k 2 - m 2 -k ε 1 - m 2 c p a 2 2π ∆u 2 2 -ε 1 - m(k + 2) 4 + k ε 1 - m 2 (1 + ln a) + k m 4 - ε 1 2 ln u 2 2 u 2 2 + c(go∆u)(t) + c 0 ,δ (go∆u) 1 1+ 0 (t).
(5.25) Then, using (3.8) and selecting m and k so small that

α 1 = k 1 - m 2 > 0, α 2 = k 2 - m 2 -k ε 1 - m 2 c p a 2 2π > 0 and α 3 = ε 1 - m(k + 2) 4 + k ε 1 - m 2 (1 + ln a) + k m 4 - ε 1 2 ln u 2 2 > 0.
Therefore, we arrive at the desired result (5.20).

Remark 5.8. Using (2.1), (2.5), (4.1), (4.6) and (4.8), we have ≤ cE(0).

E(t) = J(t) + 1 2 u t (t) 2 2 ≥ J(t) ≥ l 6 ∆u(t)
(5.27) Theorem 5.9. Let (u 0 , u 1 ) ∈ H 2 0 (Ω) × L 2 (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 .

(5.28)

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 < ∞, (5.29)
then, for any ∈ (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+ (s)ds -1 p-1+
, ∀t ≥ t 0 .

(5.30)

Proof. We multiply (5.20) by ξ(t) and use Corollary 5.4 and (5.27) to get, for any t ≥ t 0 , ξ(t)L (t) ≤ -mξ(t)E(t) + c (-E (t)) ≤ -mξ(t)E(t) + c (-E (t)) 1 (2p-1)(1+ 0 ) , ∀t ≥ t 0 .

(5.31) 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 (5.19)) 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 (5.28) with = (2p -1) 0 .

To establish (5.30), we use the idea of Messaoudi and Al-Khulaifi [START_REF] Messaoudi | General and optimal decay for a quasilinear viscoelastic equation[END_REF] , ∀t ≥ t 0 .

(5.40)

Remark 5.12. This result (k = 0) improves and generalizes many results in the literature such as Han and Wang [START_REF] Han | General decay estimate of energy for the second order evolution equations with memory[END_REF].

  multiply (3.4) by φ ∈ C ∞ 0 (0, T ) and integrate over (0, T ), we obtain, for any w ∈ V m , -s)∆u m (s)ds∆wφ(t)dxdt + T 0 Ω u m wφ(t) ln |u m | k dxdt. (3.21)

  ThenI(t) > 0, ∀t ∈ [0, T ).(4.6)

1 2p- 1 +

 11 c (-E (t)) 1 (2p-1)(1+ 0 ) ≤ -mξ(t)E(t) + c (-E (t))

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

Applying Jensen's inequality to get ξ(t)(g • ∆u)(t) ≤ η(t) 1 η(t) t 0 ξ p (s)g p (s) ∆u(t) -∆u(t -s) 2 2 ds 1 p .

Therefore, using (A2) and (5.32) we obtain

and then, according to (2.6),

So, since ξ is nonincreasing,

and hence (5.33) and (5.34) hold. Now, multiplying (5.20) by ξ(t) and using (5.27), (5.33) and (5.34) to find, for any t ≥ t 0 (as for (5.31)),

(5.35) Inequality (5.31) with 2p -1 replaced by p is exactely (5.35). Then, the proof of (5.30) can be completed as for the one of (5.28) (by taking γ = p(1 + 0 ) -1 and = p 0 ). This completes the proof of our main result.

Remark 5.10. 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 (5.17) and (5.18) drop out and, consequently, (5.20) takes the form L (t) ≤ -mE(t) + c(go∆u)(t), ∀t ≥ t 0 .

(5.36)

In this case, we obtain the following result:

Theorem 5.11. Let (u 0 , u 1 ) ∈ H 2 0 (Ω) × L 2 (Ω) and t 0 > 0. Assume that (A1) -(A2) hold. Then, there exists a positive constant K such that the solution of (1.1) satisfies, for all t ≥ t 0 , E(t) ≤ Ke -λ t t 0 ξ(s)ds if p = 1 (5.37)


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

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

C M Webler
email: cmwebler@uem.br

Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with a Kirchhoff-Carrier problem in ndimensional bounded domain under an internal finite memory. The considered class of memory kernels is very wide and allows us to derive new and optimal decay rate estimates then those ones considered previously in the literature for Kirchhoff-type models.

Introduction

The nonlinear vibrations of an elastic string are written in the form of partial integro-differential equations by

ρh ∂ 2 u ∂t 2 = p 0 + Eh 2L L 0 ∂u ∂x 2 dx ∂ 2 u ∂x 2 + f, (1.1)
for 0 < x < L and t ≥ 0, where

                            
u is the lateral deflection, x is the space coordenate variable while t denotes the time variable, E represents the Young's modulus, ρ designates the mass density, L indicates the string's lengh, h represents the cross section, p 0 denotes the axial tension, f represents an external force.

The model (1.1) has been introduced by Kirchhoff [15] in the study of the oscillations of stretched strings and plates, so that equation (1.1) is called the wave equation of Kirchhoff type until now. It is worth mentioning that, when p 0 = 0, the model (1.1) is called degenerate, and when p 0 > 0, we denominate it as a non-degenerate model.

There is a large literature regarding the Kirchhoff equation. In the sequel, we would like to mention some important works on this subject. Regarding the wellposedness of problem (1.1), the analytic case is rather known in general dimensions, as, for instance, [START_REF] Shibata | On global solvability of nonlinear viscoelastic equations in the analytic category[END_REF], [START_REF] D'ancona | Global solvability for the degenerate Kirchhoff equation with real analytic data[END_REF] and [START_REF] Pohozaev | On a class of quasilinear hyperbolic equations[END_REF]. In what concerns solutions for (1.1) lying in Sobolev spaces and, as far as we know, the results presented in the literature are only local in time, as, for example, [START_REF] Arosio | On the mildly degenerate Kirchhoff string[END_REF] and [START_REF] Menzala | On classical solutions of a quasilinear hyperbolic equation[END_REF]. However, when equation (1.1) is supplemented by some type of dissipative mechanism, which allows us, roughly speaking, to derive decay rate estimates for the solutions of the linearized problen of (1.1), it is possible to recover the global solvability in time. Consequently, deriving global solutions in time deeply depends on the decay structure of the solutions to the corresponding linearized problem of (1.1). Therefore, we are led naturally to consider the Kirchhoff equation subject to a dissipative term which guarantees the decay properties of the linearized problem. When the dissipation is given by a frictional mechanism, like g(∂ t u), there is a large body of works in the literature, see, for instance, [START_REF] Benaissa | Energy decay of solutions of a wave equation of φ-Laplacian type with a general weakly nonlinear dissipation[END_REF], [START_REF] Brito | The damped elastic strechted string equation generalized: existence, uniqueness, regularity and stability[END_REF], [START_REF] Cavalcanti | Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation[END_REF], [START_REF] Hosoda | On some nonlinear wave equations 2: global existence and energy decay of solutions[END_REF], [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF], [START_REF] Koueémon-Patcheu | Existence globale et décroissance exponentielle de l'énergy d'une équation quasilinéaire[END_REF], [START_REF] Lasiecka | Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation[END_REF], [START_REF] Menzala | On classical solutions of a quasilinear hyperbolic equation[END_REF], [START_REF] Miranda | Existence and boundary stabilization of solutions for the Kirchhoff equation[END_REF], [START_REF] Nakao | Remarks on the existence and uniqueness of global decaying of nonlinear dissipative wave equations[END_REF] and a long list of references therein.

In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with the following Kirchhoff-Carrier problem with memory:

         u ′′ -M (||∇u(t)|| 2 2 )∆u + t 0 g(t -s)∆u(s) ds = 0 in Ω × R + , u = 0 on Γ × R + , u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) in Ω , (1.2)
where Ω is a bounded domain in R n , n ∈ N * , with smooth boundary ∂Ω := Γ. While there is a great number of papers regarding the Kirchhoff equation subject to a frictional damping, in contrast, there is just a few number of papers concerned with the Kirchhoff equation subject to a dissipation given by a memory term. We are aware solely the paper [START_REF] Rivera | Existence and decay in non linear viscoelasticity[END_REF], where stronger conditions were considered on the kernel of the memory term. The assumption given in (1.7), firstly introduced in [START_REF] Messaoudi | General decay of solutions of a viscoelastic equation[END_REF], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal decay rate estimates then those ones considered previously in the literature for the linear viscoelastic wave equation. In the present paper, we combine techniques given in [START_REF] Messaoudi | General decay of solutions of a viscoelastic equation[END_REF] with new ingredients inherent to the nonlinear character of the Kirchhoff equation (1.2).

It is worth mentioning some important contributions in connection with viscoelasticity, among them, we would like to mention [START_REF] Cavalcanti | Intrinsic decay rate estimates for the wave equation with localized competing viscoelastic and frictional dissipation[END_REF], [START_REF] Cavalcanti | Frictional versus viscoelastic damping in a semilinear wave equation[END_REF], [START_REF] Dafermos | An Abstract Volterra Equation with Application to Linear Viscoelasticity[END_REF], [START_REF] Dafermos | Asymptotic Stability in Viscoelasticity[END_REF], [START_REF] Fabrizio | A new approach to equations with memory[END_REF], [START_REF] Guesmia | A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal[END_REF], [START_REF] Jiam | A Global Existence Theorem for the Dirichlet Problem in Nonlinear n-Dimensional Viscoelasticity[END_REF], [START_REF] Messaoudi | General decay of solutions of a viscoelastic equation[END_REF], [START_REF] Renardy | Mathematical Problems in Viscoelasticity[END_REF] and references therein.

∃m 0 > 0 : M ∈ C 1 (R + ) and M (λ) ≥ m 0 , ∀λ ≥ 0. (1.3) ∃γ, δ > 0 : M (λ) ≤ δλ γ , ∀λ ≥ 0. (1.4) ∃α, β > 0 : |M ′ (λ)| ≤ βλ α , ∀λ ≥ 0.
(1.5)

We shall assume the following assumptions on the kernel g:

Assumption 1.2. The function g : R + → R + belongs to the class g ∈ C 1 (R + ), g ′ ≤ 0 and, in addition

g(0) > 0 and g 0 := +∞ 0 g(s) ds < m 0 . (1.6)
Moreover, there exists a differentiable non increasing function ξ : R

+ → R * + such that ξ ′ ξ ∈ L ∞ (R + ), +∞ 0 ξ(s) ds = +∞ and g ′ (s) ≤ -ξ(s)g(s), ∀s ≥ 0. (1.7)
Now, we are in a position to state our main result.

Theorem 1.3. Assume that Assumption 1.1 and Assumption 1.2 are in place.

Then, there exists an open unbounded set S in (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) which contains (0, 0) such that, if (u 0 , u 1 ) ∈ S, and, in addition, the initial data are taken in bounded sets of

H 1 0 (Ω) × L 2 (Ω), problem (1.2) possesses a unique global solution u satisfying u ∈ L ∞ (R + ; H 2 (Ω) ∩ H 1 0 (Ω)) ∩ W 1,∞ (R + ; H 1 0 (Ω)) ∩ W 2,∞ (R + ; L 2 (Ω)). (1.8)
Furthermore, we have the following decay estimates for the energy E given in (2.10):

E(t) ≤ c E(0)e -θ t 0 ξ(s) ds , ∀t ≥ 0, (1.9)
where θ and c are positive constants independent of the initial data.

Our paper is organized as follows: in Section 2, we prove the general stability (1.9). The Section 3 is devoted to the proof the well-posedness (1.8).

D R A F T

General stability

In what follows, let us consider the Hilbert space L 2 (Ω) endowed with the inner product

(u, v) L 2 (Ω) = Ω u(x)v(x) dx
and the corresponding norm

||u|| 2 2 = Ω |u(x)| 2 dx,
and the Banach space L p (Ω), for p ≥ 1, endowed by the norm

||u|| p p = Ω |u(x)| p dx.
Let -∆ be the operator defined by the triple

H 1 0 (Ω), L 2 (Ω), ((•, •)) H 1 0 (Ω)
, where

((u, v)) H 1 0 (Ω) = Ω ∇u∇v dx, ∀u, v ∈ H 1 0 (Ω) and D(-∆) = H 2 (Ω) ∩ H 1 0 (Ω).
We recall that the Spectral Theorem for self-adjoint operators guarantees the existence of a complete orthonormal system (ω ν ) of L 2 (Ω) given by the eigenfunctions of -∆. If (λ ν ) are the corresponding eigenvalues of -∆, then

0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ ν ≤ • • • and λ ν → +∞ when ν → +∞. Moreover, ω ν √ λ ν is a complete orthonormal system in H 1 0 (Ω) and ω ν λ ν is a complete orthonormal system in H 2 (Ω) ∩ H 1 0 (Ω).
We denote by V m the subspace of

H 2 (Ω) ∩ H 1 0 (Ω) generated by the first m vectors w 1 , • • • , w m , namely, V m = [w 1 , • • • , w m ] and u m (t) = m j=1 γ jm (t)ω j , (2.1) D R A F T viscoelastic Kirchhoff-Carrier equation 5
where u m is the solution of the approximate Cauchy problem

                           (u ′′ m (t), w j ) L 2 (Ω) + M (||∇u m (t)|| 2 2 )(∇u m (t), ∇w j ) L 2 (Ω) - t 0 g(t -s)(∇u m (s), ∇w j ) L 2 (Ω) ds = 0, j = 1, • • • , m, u 0m = m j=1 γ jm (0)w j → u 0 in H 2 (Ω) ∩ H 1 0 (Ω), u 1m = m j=1 γ ′ jm (0)w j → u 1 in H 1 0 (Ω). (2.2)
By standard methods in differential equations, we can prove the existence of a solution to (2.2) on some interval [0, t m ). Then, this solution can be extended to the interval R + by using of the first estimate below.

The first estimate. Multiplying the first equation in (2.2) by γ ′ jm (t), j = 1, • • • , m, and summing the resulting expressions, we obtain

1 2 d dt ||u ′ m (t)|| 2 2 + 1 2 M (||∇u m (t)|| 2 2 ) d dt ||∇u m (t)|| 2 2 (2.3) - t 0 g(t -s)(∇u m (s), ∇u ′ m (t)) L 2 (Ω) ds = 0. Defining M (λ) = λ 0 M (s) ds, (2.4)
and since

d dt M (||∇u(t)|| 2 2 ) = d dt ||∇u(t)|| 2 2 0 M (s) ds = M (||∇u(t)|| 2 2 ) d dt ||∇u(t)|| 2 2 ,
then we deduce, taking (2.3) and the last identity into account,

1 2 d dt ||u ′ m (t)|| 2 2 + 1 2 d dt M (||∇u m (t)|| 2 2 ) - t 0 g(t -s)(∇u m (s), ∇u ′ m (t)) L 2 (Ω) ds = 0.
(2.5) Using the binary notation

(g u)(t) = t 0 g(t -s)|u(t) -u(s)| 2 ds, D R A F T
d dt Ω (g ∇u)(t)dx = Ω t 0 g ′ (t -s)|∇u(t) -∇u(s)| 2 ds dx + Ω t 0 g(t -s) d dt |∇u(t) -∇u(s)| 2 ds dx = Ω (g ′ ∇u)(t) dx +2 Ω t 0 g(t -s)(∇u(t) -∇u(s))∇u ′ (t) ds dx = Ω (g ′ ∇u)(t) dx + 2 Ω t 0 g(t -s)∇u(t)∇u ′ (t) ds dx -2 Ω t 0 g(t -s)∇u(s)∇u ′ (t) ds dx, which implies that, for u m instead of u, - t 0 g(t-s)(∇u m (s), ∇u ′ m (t)) L 2 (Ω) ds = 1 2 d dt Ω (g u m )(t)dx- 1 2 Ω (g ′ ∇u m )(t) dx - 1 2 t 0 g(s) ds d dt ||∇u m (t)|| 2 2 .
(2.6)

Then substituting (2.6) in (2.5) yields

1 2 d dt ||u ′ m (t)|| 2 2 + 1 2 d dt M (||∇um(t)|| 2 2 )+ 1 2 d dt Ω (g um)(t)dx- 1 2 t 0 g(s) ds d dt ||∇um(t)|| 2 2 = 1 2 Ω (g ′ ∇u m )(t) dx,
and using 1 2

d dt t 0 g(s) ds ||∇u m (t)|| 2 2 = 1 2 g(t)||∇u m (t)|| 2 2 + 1 2 t 0 g(s) ds d dt ||∇u m (t)|| 2 2 ,
we get

1 2 d dt ||u ′ m (t)|| 2 2 + M (||∇u m (t)|| 2 2 )+ Ω (g ∇u m )(t)dx - t 0 g(s) ds ||∇u m (t)|| 2 2 = 1 2 Ω (g ′ ∇u m )(t) dx - 1 2 g(t)||∇u m (t)|| 2 2 .
(2.7)

On the other hand, the hypothesis (1.3) implies that

M ||∇u m (t)|| 2 2 = ||∇um(t)|| 2 2 0 M (s) ds ≥ m 0 ||∇u m (t)|| 2 2 , D R A F T viscoelastic Kirchhoff-Carrier equation
M ||∇u m (t)|| 2 2 - t 0 g(s) ds ||∇u m (t)|| 2 2 ≥ (m 0 -g 0 )||∇u m (t)|| 2 2 .
(2.8)

Combining (2.7) and (2.8), and observing that g > 0 and g ′ ≤ 0, we deduce

1 2 ||u ′ m (t)|| 2 2 + 1 2 (m 0 -g 0 )||∇u m (t)|| 2 2 + Ω (g ∇u m )(t)dx (2.9) ≤ 1 2 ||u 1m || 2 2 + 1 2 M (||∇u 0m || 2 2 ) ≤ L 1 ||u 1 || 2 2 , ||∇u 0 || 2 2 , ∀t ≥ 0, ∀m ∈ N,
where L 1 does not depend neither on m ∈ N nor on t ≥ 0. This implies that the approximated solution u m exists globally in the topologies given in (2.9).

Defining the energy E associated to problem (1.2) by

E(t) := 1 2 ||u ′ (t)|| 2 2 + 1 2 M (||∇u(t)|| 2 2 ) - 1 2 t 0 g(s) ds ||∇u(t)|| 2 2 (2.10) + 1 2 Ω (g ∇u)(t)dx,
then, in view of (2.7), it is non increasing function. In addition, as a consequence of (2.7), the following identity of the energy holds:

E(t 2 ) -E(t 1 ) = 1 2 t2 t1 Ω g ′ ✷∇u -g(t)|∇u| 2 dxdt ≤ 0, ∀t 2 ≥ t 1 ≥ 0. (2.11)
Energy decay estimate. Define

(g • v)(t) = t 0 g(t -s)||v(t) -v(s)|| 2 2 ds
and

(g ⋄ v)(t) = t 0 g(t -s)(v(t) -v(s)) ds. Lemma 2.1. Let ψ ∈ L 1 (R + , R + ) and u ∈ L 2 (R + ; L 2 (Ω)). Then ||(ψ ⋄ u)(t)|| 2 2 ≤ ||ψ|| L 1 (R+) (ψ • u)(t).

Guesmia, Messaoudi and Webler

Proof. Applying Hölder inequality and Fubini theorem, we have

||(ψ ⋄ u)(t)|| 2 2 = Ω t 0 ψ(t -s) ψ(t -s)(u(t) -u(s)) ds 2 dx ≤ t 0 ψ(ζ) dζ t 0 ψ(t -s) Ω (u(t) -u(s)) 2 dx ds.
✷ From now on, for short notation, we shall drop the parameter "m" in u m . We have the following useful lemma: Lemma 2.2. Let u be a solution to the approximated problem (2.2) corresponding to initial data taken in bounded sets of H 1 0 (Ω)×L 2 (Ω). Then, we have the following decay rate estimate:

E(t) ≤ c E(0)e -θ t 0 ξ(s) ds , t ≥ 0,
for some positive constants c and θ which do not depend on m ∈ N.

Proof. From (2.2), we have,

(u ′′ (t), w) L 2 (Ω) + M (||∇u(t)|| 2 2 )(∇u(t), ∇w) L 2 (Ω) (2.12) - t 0 g(t -s)(∇u(s), ∇w) L 2 (Ω) ds = 0, ∀w ∈ V m .
Recovering the potential energy.

Substituting w = u in (2.12), multiplying by ξ(t) and integrating over [0, T ], we can write

T 0 ξ(t)(u ′′ (t), u(t)) L 2 (Ω) dt + T 0 ξ(t)M (||∇u(t)|| 2 2 )||∇u(t)|| 2 2 dt (2.13) - T 0 ξ(t) t 0 g(t -s)(∇u(s), ∇u(t)) L 2 (Ω) ds dt = 0.
Having in mind that

d dt ξ(t)(u ′ (t), u(t)) L 2 (Ω) = ξ(t)(u ′′ (t), u(t)) + ξ(t)||u ′ (t)|| 2 2 + ξ ′ (t)(u ′ (t), u(t)) L 2 (Ω) ,
from (2.13) we obtain and, using (1.3) from (2.14), we find

ξ(t)(u ′ (t), u(t)) L 2 (Ω) | T 0 - T 0 ξ(t)||u ′ (t)|| 2 2 dt - T 0 ξ ′ (t)(u ′ (t), u(t)) L 2 (Ω) dt + T 0 ξ(t)M (||∇u(t)|| 2 2 )||∇u(t)|| 2 2 dt (2.14) - T 0 ξ(t) t 0 g(t -s)(∇u(s), ∇u(t)) L 2 (Ω) ds dt = 0, D R A F T viscoelastic Kirchhoff-Carrier equation
m 0 T 0 ξ(t)||∇u(t)|| 2 2 dt ≤ -ξ(t)(u ′ (t), u(t)) L 2 (Ω) | T 0 (2.15) + T 0 ξ(t)||u ′ (t)|| 2 2 dt + T 0 ξ ′ (t)(u ′ (t), u(t)) L 2 (Ω) dt + T 0 ξ(t) t 0 g(t -s)(∇u(s), ∇u(t)) L 2 (Ω) ds dt.
Now, we will estimate separately the last terms on the right hand side of (2.15). We have, using Cauchy-Schwarz and Young's inequalities,

T 0 ξ(t) t 0 g(t -s)(∇u(s), ∇u(t)) L 2 (Ω) ds dt ≤ T 0 ξ(t) t 0 g(t -s)||∇u(s)|| 2 ||∇u(t)|| 2 ds dt ≤ T 0 ξ(t) t 0 g(t -s) (||∇u(s) -∇u(t)|| 2 + ||∇u(t)|| 2 ) ||∇u(t)|| 2 ds dt = T 0 ξ(t) t 0 g(t -s)||∇u(s) -∇u(t)|| 2 ||∇u(t)|| 2 ds dt + T 0 ξ(t) t 0 g(t -s)||∇u(t)|| 2 2 ds dt ≤ (1 + ε) T 0 ξ(t) t 0 g(t -s)||∇u(t)|| 2 2 ds dt + 1 4ε T 0 ξ(t)(g • ∇u)(t) dt; that is, T 0 ξ(t) t 0 g(t -s)(∇u(s), ∇u(t)) L 2 (Ω) ds dt ≤ (1 + ε)g 0 T 0 ξ(t)||∇u(t)|| 2 2 dt (2.16) + 1 4ε T 0 ξ(t)(g • ∇u)(t) dt.
On the other hand, because ξ ′ ξ is bounded, we see that, for any ǫ 0 > 0,

T 0 ξ ′ (t)(u ′ (t), u(t)) L 2 (Ω) dt ≤ c 0 T 0 ξ(t) ǫ 0 ||u ′ (t)|| 2 2 + 1 ǫ 0 ||∇u(t)|| 2 2 dt, (2.17)
where

c 0 = 1 2 (1 + λ -1/2 1 )|| ξ ′ ξ || L ∞ (R+) . D R A F T 10
Guesmia, Messaoudi and Webler From (2.15), (2.16) and (2.17) we arrive at

m 0 T 0 ξ(t)||∇u(t)|| 2 2 dt ≤ -ξ(t)(u ′ (t), u(t)) L 2 (Ω) | T 0 + (1 + ǫ 0 c 0 ) T 0 ξ(t)||u ′ (t)|| 2 2 dt (2.18) + (1 + ε)g 0 + c 0 ǫ 0 T 0 ξ(t)||∇u(t)|| 2 2 dt + 1 4ε T 0 ξ(t)(g • ∇u)(t) dt.
Recovering the kinectic energy. Substituting w = g ⋄ u ∈ V m in (2.12) and multiplying by ξ(t), it results that

T 0 ξ(t)(u ′′ (t), (g ⋄ u)(t)) L 2 (Ω) dt (2.19)
+ T 0 ξ(t)M ||∇u(t)|| 2 2 (∇u(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt - T 0 ξ(t) t 0 g(t -s)(∇u(s), (g ⋄ ∇u)(t)) L 2 (Ω) ds dt = 0. But d dt ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) = ξ(t)(u ′′ (t), (g ⋄ u)(t)) L 2 (Ω) +ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) L 2 (Ω) +ξ(t) u ′ (t), t 0 g(t -s)u ′ (t) ds L 2 (Ω) +ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) .
Integrating the last identity over (0, T ), we obtain,

T 0 ξ(t)(u ′′ (t), (g ⋄ u)(t)) L 2 (Ω) dt = ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) | T 0 (2.20) - T 0 ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) dt - T 0 ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) L 2 (Ω) dt - T 0 ξ(t) t 0 g(s) ds ||u ′ (t)|| 2 2 dt. D R A F T viscoelastic Kirchhoff-
= ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) | T 0 - T 0 ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) + T 0 ξ(t)M ||∇u(t)|| 2 2 (∇u(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt - T 0 ξ(t) t 0 g(t -s) (∇u(s), (g ⋄ ∇u)(t)) L 2 (Ω) ds dt - T 0 ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) dt.
Let t 0 > 0 such that g(t 0 )t 0 > 0. This is possible in vertue of Assumption 1.2. Then one has t 0 g(s) ds ≥ g(t 0 )t 0 > 0, ∀t ≥ t 0 .

(

≤ ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) | T 0 - T 0 ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) L 2 (Ω) dt + T 0 ξ(t)M ||∇u(t)|| 2 2 (∇u(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt - T 0 ξ(t) t 0 g(t -s) (∇u(s), (g ⋄ ∇u)(t)) L 2 (Ω) ds dt - T 0 ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) dt, ∀T ≥ t 0 .
On the other hand, it is convenient to observe that

T 0 ξ(t) t 0 g(t -s)∇u(t) ds, (g ⋄ ∇u)(t) L 2 (Ω) dt (2.24) = T 0 ξ(t) ((g ⋄ ∇u)(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt + T 0 ξ(t) t 0 g(t -s)∇u(s) ds, (g ⋄ ∇u)(t) L 2 (Ω)
dt.

D R A F T 12

Guesmia, Messaoudi and Webler

Combining (2.23) and (2.24) we infer, for all T ≥ t 0 ,

g(t 0 )t 0 T t0 ξ(t)||u ′ (t)|| 2 2 dt ≤ ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) | T 0 + T 0 ξ(t)||(g ⋄ ∇u)(t)|| 2 2 dt - T 0 ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) L 2 (Ω) dt + T 0 ξ(t)M ||∇u(t)|| 2 2 (∇u(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt - T 0 ξ(t) t 0 g(t -s)∇u(t) ds, (g ⋄ ∇u)(t) L 2 (Ω) dt - T 0 ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) dt. (2.25)
Next, we shall analyse the terms on the right hand side of (2.25).

Estimate for

I 1 := ξ(t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) | T 0 .
We have,

I 1 = ξ(T ) u ′ (T ), T 0 g(T -s)(u(T ) -u(s))ds L 2 (Ω)
.

(2.26) Thus, having in mind lemma 2.1, the definition of the energy in (2.10) and that ξ is non increasing, we deduce

|I 1 | = ξ(T )| T 0 g(T -s)(u ′ (T ), u(T ) -u(s)) L 2 (Ω) ds| (2.27) ≤ ξ(0) T 0 g(T -s)||u ′ (T )|| 2 ||u(T ) -u(s)|| 2 ds ≤ ξ(0) T 0 g(T -s) 1 2 ||u ′ (T )|| 2 2 + 1 2 ||u(T ) -u(s)|| 2 2 ds ≤ 1 2 ξ(0)g 0 ||u ′ (T )|| 2 2 + λ -1/2 1 ξ(0) 2 T 0 g(T -s)||∇u(T ) -∇u(s)|| 2 2 ds = 1 2 ξ(0)g 0 ||u ′ (T )|| 2 2 + λ -1/2 1 ξ(0) 2 (g • ∇u)(T ) ≤ ξ(0) g 0 + λ -1/2 1 E(T ).
Therefore for some C > 0, which, from now on, will represent various constants do not depend on T and m ∈ N, which is crucial in the proof.

|I 1 | ≤ C E(T ), ( 2
Estimate for

I 2 := - T 0 ξ(t)(u ′ (t), (g ′ ⋄ u)(t)) L 2 (Ω) dt.
Employing lemma 2.1 and the property ξ(t) ≤ ξ(0), one has

|I 2 | ≤ T 0 ξ(t)||u ′ (t)|| 2 ||(g ′ ⋄ u)(t)|| 2 dt (2.29) ≤ ε T 0 ξ(t)||u ′ (t)|| 2 2 dt + 1 4ε T 0 ξ(t)||(g ′ ⋄ u)(t)|| 2 2 dt ≤ ε T 0 ξ(t)||u ′ (t)|| 2 2 dt + ξ(0) 4ε ||g ′ || L 1 (R+) T 0 (|g ′ | • u)(t)dt ≤ ε T 0 ξ(t)||u ′ (t)|| 2 dt - ξ(0)λ -1/2 1 4ε ||g ′ || L 1 (R+) T 0 (g ′ • ∇u)(t) dt,
where ε is an arbitrary positive constant.

Similarly, because ξ ′ ξ is bounded, we have

| - T 0 ξ ′ (t)(u ′ (t), (g ⋄ u)(t)) L 2 (Ω) dt| ≤ ε T 0 ξ(t)||u ′ (t)|| 2 dt (2.30) + λ -1/2 1 g 0 4ε || ξ ′ ξ || L ∞ (R+) T 0 ξ(t)(g • ∇u)(t)dt,
where ε is an arbitrary positive constant.

Estimate for

I 3 := T 0 ξ(t)M ||∇u(t)|| 2 2 (∇u(t), (g ⋄ ∇u)(t)) L 2 (Ω) dt.
Let us define:

E(t) := 1 2 ||u ′ (t)|| 2 2 + 1 2 ||∇u(t)|| 2 2 , (2.31)
the mechanical energy associated to problem (1.2). First, we observe that

E(t) ≥ 1 2 ||u ′ (t)|| 2 2 + (m 0 -g 0 )||∇u(t)|| 2 2 , which implies that E(0) ≥ E(t) ≥ α 0 E(t), ∀t ≥ 0,
where α 0 = min{1, m 0g 0 } (α 0 > 0 in vertue of (1.6)). Thus, we get

E(t) ≤ α -1 0 E(t) ≤ α -1 0 E(0), ∀t ≥ 0. (2.32) D R A F T 14
Guesmia, Messaoudi and Webler

Using the assumption (1.4) and taking (2.32) into account, we deduce, similarly to the estimate (2.29),

|I 3 | ≤ δα -γ 0 (2 E(0)) γ T 0 ξ(t)||∇u(t)|| 2 ||(g ⋄ ∇u)(t)|| 2 dt (2.33) ≤ ε T 0 ξ(t)||∇u(t)|| 2 2 dt + δ 2 α -2γ 0 (2 E(0)) 2γ 4ε T 0 ξ(t)||(g ⋄ ∇u)(t)|| 2 2 dt ≤ ε T 0 ξ(t)||∇u(t)|| 2 2 dt + δ 2 α -2γ 0 g 0 (2 E(0)) 2γ 4ε T 0 ξ(t)(g • ∇u)(t)dt,
where ε is an arbitrary positive constant.

Estimate for

I 4 := T 0 ξ(t)||(g ⋄ ∇u)(t)|| 2 2 . Lemma 2.1 implies that |I 4 | ≤ g 0 T 0 ξ(t)(g • ∇u)(t) dt. (2.34)
Estimate for

I 5 := - T 0 ξ(t) t 0 g(t -s)∇u(t) ds, (g ⋄ ∇u)(t) L 2 (Ω)
dt. One has, using again lemma 2.1,

|I 5 | ≤ g 0 T 0 ξ(t)||∇u(t)|| 2 ||(g ⋄ ∇u)(t)|| 2 dt (2.35) ≤ ε T 0 ξ(t)||∇u(t)|| 2 2 dt + g 2 0 4ε T 0 ξ(t)||(g ⋄ ∇u)(t)|| 2 2 dt ≤ ε T 0 ξ(t)||∇u(t)|| 2 2 dt + g 3 0 4ε T 0 ξ(t)(g • ∇u)(t)dt.
Combining (2.25), (2.28), (2.29), (2.30), (2.33), (2.34) and (2.35), we conclude, for all T ≥ t 0 ,

g(t 0 )t 0 T t0 ξ(t)||u ′ (t)|| 2 2 dt ≤ 2ε T 0 ξ(t)||u ′ (t)|| 2 dt + 2ε T 0 ξ(t)||∇u(t)|| 2 2 dt (2.36) +C E(T ) + C T 0 (ξ(t)(g • ∇u)(t) -(g ′ • ∇u)(t)) dt.
Multiplying (2.18) by a constant β 1 > 0, adding (2.36) and having in mind that, according to the properity ξ(t) ≤ ξ(0) and (2.32), and

g(t 0 )t 0 t 0 ξ(t)||u ′ (t)|| 2 2 dt ≤ g(t 0 )t 0 ξ(0) t0 0 ||u ′ (t)|| 2 2 dt ≤ C E(0), ∀t ∈ [0, t 0 ] D R A F T viscoelastic Kirchhoff-Carrier equation
| -ξ(t)(u ′ (t), u(t)) L 2 (Ω) | T 0 | ≤ C E(0), ∀t ≥ 0,
we can write

(g(t 0 )t 0 -2ε -β 1 (1 + ǫ 0 c 0 )) T 0 ξ(t)||u ′ (t)|| 2 2 dt
(2.37)

+ β 1 m 0 -g 0 - c 0 ǫ 0 -ε(β 1 g 0 + 2) T 0 ξ(t)||∇u(t)|| 2 2 dt ≤ C E(0) + C T 0 (ξ(t)(g • ∇u)(t) -(g ′ • ∇u)(t))dt, ∀T ≥ t 0 . Choosing ǫ 0 > c0 m0-g0 , 0 < β 1 < g(t0)t0 1+ǫ0c0 and 0 < ε < min 1 2 (g(t 0 )t 0 -β 1 (1 + ǫ 0 c 0 )), β 1 (m0-g0- c 0 ǫ 0 ) β 1 g0+2
. Hence, from (2.37), we deduce

T 0 ξ(t)||ξ(t)u ′ (t)|| 2 2 dt + T 0 ξ(t)||∇u(t)||
E(t) = 1 2 ||u ′ (t)|| 2 2 + M (||∇u(t)|| 2 2 ) + (g • ∇u)(t) - t 0 g(s) ds ||∇u(t)|| 2 2 ≤ 1 2 g • ∇u + 1 2 ||u ′ (t)|| 2 2 + δ γ + 1 ||∇u(t)|| 2γ 2 ||∇u(t)|| 2 2 (2.41) ≤ 1 2 (g • ∇u)(t) + 1 2 ||u ′ (t)|| 2 2 + δ γ + 1 2 α 0 E(0) γ ||∇u(t)|| 2 2 .
We are assuming, by assumption, that the initial data are taken in bounded sets of H nor on t ∈ R + ) such that E(t) ≤ L. This implies that there exists d > 0 such that E(0) < d. Then, from (2.41), we conclude

E(t) ≤ 1 2 (g • ∇u)(t) + B 0 E(t), ∀t ≥ 0, (2.42)
where B 0 = max 1, δ(2d) γ (γ + 1)α γ 0 , and, therefore

T 0 ξ(t) E(t) dt ≤ 1 2 T 0 ξ(t)(g • ∇u)(t) dt + B 0 T 0 ξ(t)E(t)
E ′ (t) ≤ 1 2 (g ′ • ∇u)(t), ∀t ≥ 0, it implies that (-g ′ • ∇u)(t) ≤ -2 E ′ (t), ∀t ≥ 0,
and consequently, from (2.44), we have

T 0 ξ(t) E(t) dt ≤ C E(0) + C T 0 ξ(t)(g • ∇u)(t) dt -C T 0 E ′ (t) dt, ∀T ≥ t 0 , namely, T 0 ξ(t) E(t) dt ≤ C E(0) + C T 0 ξ(t)(g • ∇u)(t) dt, ∀T ≥ t 0 . (2.45)
Once we are assuming (1.7) and because ξ is non increasing, we see that

ξ(t)(g • ∇u)(t) ≤ ((ξg) • ∇u)(t) ≤ -(g ′ • ∇u)(t) ≤ -2 E ′ (t),
then, we deduce from (2.45) that

T 0 ξ(t) E(t) dt ≤ C E(0) -C T 0 E ′ (t), ∀T ≥ t 0 , ( 2
F (t) dt ≤ CF (0), ∀T ≥ 0.
Consequently, by applying Theorem 9.1 in [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF], we find that ther exist positive constants c and θ not depending on Ê(0) such that

F (t) ≤ cF (0)e -θt , ∀t ≥ 0.
✷ By the definition of F , this last inequality implies the general stability (1.9), which finishes the proof.

Well-posedness

Lemma 3.1 (H 2 (Ω) a priori bounds). Suppose that u is a local solution on [0, T [ such that sup t∈[0,T [ {||∇u ′ (t)|| 2 , ||∆u(t)|| 2 } < K,
for some K > 0 and T > 0. Then, the following estimate holds:

||∇u ′ (t)|| 2 2 + ||∆u(t)|| 2 2 ≤ CK 3 ( E(0)) 2α+1 2 t 0 e -θ(2α+1) 2 s 0 ξ(τ ) dτ ds (3.1) +α -1 0 ||∇u 1 || 2 2 + M (||∇u 0 || 2 2 )||∆u 0 || 2 2 := G(t, I 0 , I 1 , K) on [0, T [, with I 0 = E(0) and I 1 = ||∇u 1 || 2 2 + M (||∇u 0 || 2 2 )||∆u 0 || 2 2 .
Proof. Taking w = -∆u ′ ∈ V m in the approximate problem (2.2) yields

1 2 d dt ||∇u ′ (t)|| 2 2 + M (||∇u(t)|| 2 2 )||∆u(t)|| 2 2 - t 0 g(t -s)(∆u(s), ∆u ′ (s)) L 2 (Ω) ds = M ′ (||∇u(t)|| 2 2 )(∇u ′ (t), ∇u(t)) L 2 (Ω) ||∆u(t)|| 2 2 . (3.2)
Guesmia, Messaoudi and Webler

Considering similar computations as done before, from (3.2), we infer

1 2 d dt ||∇u ′ (t)|| 2 2 +M (||∇u(t)|| 2 2 )||∆u(t)|| 2 2 - t 0 g(s)ds ||∆u(t)|| 2 2 +(g • ∆u)(t) = 1 2 (g ′ • ∆u)(t) dx - 1 2 g(t)||∆u(t)|| 2 2 + M ′ (||∇u|| 2 2 )(∇u ′ (t), ∇u(t)) L 2 (Ω) ||∆u(t)|| 2 2 . (3.3) Integrating (3.3) over (0, t), t > 0, we deduce ||∇u ′ (t)|| 2 2 + M ||∇u(t)|| 2 2 ||∆u(t)|| 2 2 - t 0 g(s)ds ||∆u(t)|| 2 2 + (g • ∆u)(t) -||∇u 1 || 2 2 + M ||∇u 0 || 2 2 ||∆u 0 || 2 2 (3.4) ≤ 2 t 0 M ′ ||∇u(s)|| 2 2 (∇u ′ (s), ∇u(s)) L 2 (Ω) ||∆u(s)|| 2 2 ds.
On the other hand, we have, in vertue of (1.3) of and (1.6), 2 are satisfied, for example, if g converges to zero at infinity faster than 1 t d , for any d > 0, like g 1 (t) = a 1 e -b1(t+1) q 1 and g 2 (t) = a 2 e -b2(ln(t+e q 2 -1 )) q 2 , where a i , b i , q 1 > 0 and q 2 > 1 such that a i are small enough so that (1.6) holds. For these two particular examples, ξ is given, respectively, by ξ(t) = b 1 q 1 (t + 1) min{0,q1-1} and ξ(t) = b 2 q 2 (t + e q2-1 ) -1 (ln(t + e q2-1 )) q2-1 .

||∇u ′ (t)|| 2 2 + M ||∇u(t)|| 2 2 ||∆u(t)|| 2 2 - t 0 g(s)ds ||∆u(t)|| 2 2 + (g • ∆u)(t) dx ≥ ||∇u ′ (t)|| 2 2 + (m 0 -g 0 )||∆u(t)|| 2 2 (3.5) ≥ α 0 ||∇u ′ (t)||
Howover, when g converges to zero at infinity slower than 1 t d , for some d > 0, like g 3 (t) = a 3 (t + 1) -q3 , where a 3 > 0 and q 3 > 1, Assumption 1.2 is satisfied with ξ(t) = q 3 (t + 1) -1

provided that a 3 is small enough so that (1.6) holds. But (3.8) is not always satisfied, since (3. Recalling Lemma 2.2, one can assert that u (the approximate solution constructed by Galerkin method) and u ′ exist globally in R + . Suppose that (u 0 , u 1 ) ∈ S K for some K > 0. Thus, we would like to prove that ||∆u(t)|| 2 < K and ||∇u ′ (t)|| 2 < K, ∀t ≥ 0.

1

  Typeset by B S P M style. c Soc. Paran. de Mat.

  The following assumptions are made on the function M :

4 Guesmia,

 4 Messaoudi and Webler

6 Guesmia

 6

  (1.6) into account,

  1 0 (Ω) × L 2 (Ω). Consequently, let L > 0 (not depending neither on m ∈ N

. 8 ) 3 . 4 .

 834 τ ) dτ ds < +∞. (3Remark Condition (3.8) as well as Assumption 1.

8 ) is equivalent to 1 2 (

 82 2α + 1)θq 3 > 1.Assume that Assumption 1.1, Assumption 1.2 and (3.8) hold, let K > 0 and setS K := {(u 0 , u 1 ) ∈ (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω), G(t, I 0 , I 1 , K) < K 2 , ∀t ≥ 0},

(3. 11 )

 11 In order to prove(3.11), we argue by contradiction. So, assume that (3.11) does not hold. Then, there exists some T > 0 such that||∆u(t)|| 2 < K and ||∇u ′ (t)|| 2 < K, ∀t ∈ [0, T [ (3.12) and ||∆u(T )|| 2 = K or ||∇u ′ (T )|| 2 = K. (3.13)

  dt, ∀T ≥ 0. (2.43) Combining (2.39) and (2.43) we deduce, for all T ≥ t 0 ,

	0	T	ξ(t) E(t) dt ≤ C E(0) + C	0	T	(ξ(t)(g • ∇u)(t) -(g ′ • ∇u)(t)) dt.	(2.44)
	Since, according to (2.7),

Acknowledgments

This work was initiated during the visit of the first author to the State University of Maringá in March 2014. The first author wishes to thank this university for its kind support and hospitality.

D R A F T 20

Guesmia, Messaoudi and Webler Repeating the proof of Lemma 3.1, we see from (3.12) and (3.13) that (3.1) remains valid, for 0 ≤ t < T , so that, taking (3.9) into account, one has

which contradicts (3.13). Thus, we have shown (3.11). As a consequence, we can repeat the continuation procedure indefinitely and we can conclude that, if (u 0 , u 1 ) ∈ S, the solution u can be continued globally on R + and (u(t), u ′ (t)) ∈ S, for all t ≥ 0.

Uniqueness. Let u and v be two solutions to problem (1.2). Then

Taking the inner product in L 2 (Ω) of the first equation of the above system with w ′ , we deduce