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

. In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han and Wang [26]. 35B35, 35L55, 75D05,

1. Introduction. In this paper, we deal with the existence and decay of solutions of the following plate problem: 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 [31] 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. [43] 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 [29] investigated the energy decay of a plate model under weak growth assumptions on the feedback function. Messaoudi [36] 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 [13].
1.2. Viscoelastic problems. Since the pioneer works of Dafermos [15,16] 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 [14]. Hrusa [28] considered a one-dimensional nonlinear viscoelastic equation of the form 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 [17], Dassios and Zafiropoulos considered a viscoelastic problem in R 3 and proved a polynomial decay result for exponentially decaying kernels. After that, Rivera [42] 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. [7] studied, in a bounded domain, the following equation: 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 [39,40], 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 [8], Cavalcanti et al. considered 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 , and established an exponential decay result under some restrictions on ω. Berrimi and Messaoudi [4] established the result of [8], under weaker conditions on both a and g, to a problem where a source term is competing with the damping term. Cavalcanti and Oquendo [10] considered the following problem: 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. [34] 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 [37,38] considered 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. [24] studied the well-posedness and stability of the following coupled two wave equations: 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 [37,38] with ξ = min{ξ 1 , ξ 2 }. Very recently, Messaoudi and Al-Khulaifi [41] 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 [9,23] 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 [1,18]. This type of problems has applications in many branches of physics such as nuclear physics, optics and geophysics [2,5,19]. Birula and Mycielski [5,6] studied the following problem: 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 [20,45]. In [11], Cazenave and Haraux considered and established the existence and uniqueness of the solution for the Cauchy problem. Gorka [19] 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 [2] proved the existence of classical solutions and investigated the weak solutions for the corresponding onedimensional Cauchy problem for equation (1.9). Hiramatsu et al. [27] introduced the following equation: 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 [25], 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.

2.
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.

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(A2) There exist a nonincreasing differentiable function ξ : and c p is the smallest positive number satisfying s is a continuous and decreasing function on (0, ∞), with Then, there exists a unique k 0 > 0 such that f (k 0 ) = 0. Moreover, The energy functional associated with problem (1.1) is Direct differentiation of (2.5), using (1.1), leads to 21,12], Logarithmic Sobolev inequality). Let u be any function in H 1 0 (Ω) and a be any positive real number. Then Corollary 2.3. Let u be any function in H 2 0 (Ω) and a be any positive real number. Then 3. Local existence. In this section, we state and prove the local existence result for problem (1.1).
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 .., w m } and let the projections of the initial data on the finite dimensional subspace V m be given by We search for an approximate solution of the approximate problem in V m : 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 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 . The last inequality together with (2.1) and the Logarithmic Sobolev inequality (2.8) lead to This selection is possible thanks to (A3). So, we get Then, using Cauchy-Schwarz' inequality, we get hence, inequality (3.9) gives Without loss of generality, we take C 1 ≥ 1, which gives Applying the Logarithmic Gronwall inequality to the last inequality, we obtain the following estimate: Hence, from inequality (3.9) it follows that: 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 (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 , (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, 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

EXISTENCE AND A GENERAL DECAY FOR A VISCOELASTIC PLATE EQUATION 167
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), 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)).
On the other hand, using Hölder's inequality and (4.4), we find which implies the left inequality in (4.3).

EXISTENCE AND A GENERAL DECAY FOR A VISCOELASTIC PLATE EQUATION 169
Proof. From (4.2), we have Substitute (4.7) in (4.1), we find We assume by contradiction that t 0 ∈ (0, T ). Thus I(t 0 ) = 0 and 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 By using (4.9) and Lemma 4.1, we have 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 The global existence can be easily established by repeating the steps of the proof of Theorem 3.1 [40]. 5. 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.

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

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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, then, puting δ 4 = cδ 1 and using Hölder's inequality and Lemma 5.1, we find (5.18) The above inequalities imply (5.11).
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 Using the Logarithmic Sobolev inequality, for 0 < m < 2ε 1 , we get (5.23) At this point we choose δ so small that Whence δ is fixed, the choice of any two positive constants ε 1 and ε 2 satisfying will make Then, we choose ε 1 and ε 2 so small so that (5.21) and (5.24) remain valid and, further,

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Consequently, we get (5.19) and (5.25) Then, using (3.8) and selecting m and k so small that Therefore, we arrive at the desired result (5.20).

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Applying Jensen's inequality to get So, since ξ is nonincreasing,  (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.