Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity

In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$h′(t)≤−ξ(t)H(h(t)),t≥0, where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.


Introduction
In this work, we consider the following viscoelastic plate problem with velocity-dependent material density and logarithmic nonlinearity: equipped with initial and boundary conditions where Ω is a bounded domain of R 2 with smooth boundary ∂Ω, n is the unit outer normal to ∂Ω, and ρ and α are positive constants. The relaxation function h satisfies the following general condition: where the functions ξ and H satisfy some conditions specified later. To motivate our work, let us recall some results regarding problems with logarithmic nonlinearity.

Problems with logarithmic nonlinearity
The logarithmic nonlinearity has many applications in physics such as nuclear physics, optics, and geophysics [1][2][3]. For the problems with logarithmic nonlinearity, we start with the works of Birula and Mycielski [4] and [5], where they proved that the wave equations with logarithmic nonlinearity have stable and localized solutions. Cazenave and Haraux [6] considered the Cauchy problem u ttu = u ln |u| α (4) in R 3 and established the existence and uniqueness of the solution. The corresponding one-dimensional problem of (4) was studied by Gorka [1], who established the global existence of weak solutions, provided that (u 0 , u 1 ) ∈ H 1 0 × L 2 . Bartkowski and Gorka [2] investigated weak solutions and also proved the existence of classical solutions. Hiramatsu et al. [3] considered the problem u ttu + u + u t + |u| 2 u = u ln |u| (5) and investigated numerical solutions of this problem without theoretical analysis. Recently, Al-Gharabli et al. [7] considered the problem and proved existence and decay results of the solutions under the following condition on the relaxation function: Al-Gharabli et al. [8] considered the problem and as in [7] proved the existence and decay results for the solutions with imposing the same condition (7). Very recently, Al-Gharabli [9] considered the same problem (6) and established a general decay result for which the relaxation function h satisfies h (t) ≤ -ξ (t)H(h(t)). For more results on some problems with logarithmic nonlinearity, we refer to the recent works [10][11][12][13][14].

Problems with infinite memory
Giorgi et al. [15] considered the following semilinear hyperbolic equation with linear memory in a bounded domain Ω ⊂ R 3 : with K(0), K(+∞) > 0 and K ≤ 0 and proved the existence of global attractors for the solutions. Conti and Pata [16] considered the following semilinear hyperbolic equation: where the memory kernel is a convex decreasing smooth function such that K(0) > K(+∞) > 0, and g : R + → R + is a nonlinear term of at most cubic growth satisfying some conditions. They proved the existence of a regular global attractor. Appleby et al. [17] studied the linear integro-differential equation and established an exponential decay result for strong solutions in a Hilbert space. Pata [18] discussed the decay properties of the semigroup generated by the following equation: where A is a strictly positive self-adjoint linear operator, α > 0, β ≥ 0, and the memory kernel μ is a decreasing function satisfying specific conditions. Subsequently, they established necessary and sufficient conditions for the exponential stability. Guesmia [19] considered the equation and introduced a new ingenuous approach for proving a more general decay result based on the properties of convex functions and the generalized Young inequality. He used a larger class of infinite history kernels satisfying the condition with where H : R + → R + is an increasing strictly convex function. Using this approach, Guesmia and Messaoudi [20] later considered the equation in a bounded domain under suitable conditions on a 1 and a 2 for a wide class of relaxation functions h 1 and h 2 , which are not necessarily decaying polynomially or exponentially, and established a general decay result such that the usual exponential and polynomial decay rates are only particular cases. Messaoudi and Al-Gharabli [7] considered the nonlinear wave equation in which the relaxation function g satisfies and they proved a general decay result on the solution energy using an approach different from that introduced by Guesmia [19]. Recently, Al-Mahdi and Al-Gharabli [21] considered the viscoelastic problem established decay results in which the relaxation function h satisfies and obtained a better decay rate than that in [19] and [22]. For more results on problems with infinite memory and finite memory, we refer the reader to [23][24][25][26][27]. Motivated by all these works, we intend to establish a three-fold objective: (a) To extend many earlier works for the wave equations such as those discussed in [1,3,7,[28][29][30] to the plate equation with logarithmic nonlinearity. (b) To extend some general decay results, known for the case of finite history, to the case of infinite history where the relaxation function satisfies a wider class of relaxation functions instead of those considered in [7,8,12,19,21,29,31]. (c) To drop the boundedness assumptions on the history data considered in many earlier results in [7,19,21]. We obtain our results by using the multiplier method with some logarithmic inequalities and some properties of integro-differential equations and inequalities. Our decay result is based on ξ , H, and α. This paper is organized as follows. In Sect. 2, we present some notations, assumptions, and a local and global existence result of our problem. In Sect. 3, we establish some lemmas needed in the proof of our result. Stability results with an example are presented in Sect. 4. Some conclusions are given in Sect. 5.

Preliminaries
In this section, we introduce our assumptions and give some useful lemmas. We use c to denote a positive generic constant.

Remark 2.2 If
H is a strictly increasing and strictly convex C 2 function on (0, r] with H(0) = H (0) = 0, then it has an extension H that is strictly increasing and strictly con- For simplicity, in the rest of this paper, we use H instead of H.
Since H is strictly convex on (0, r] and H(0) = 0, then Remark 2.4 The function g(s) = 2π c p se -3 2 is a continuous decreasing function on (0, ∞) with Then there exists a unique α 0 > 0 such that g(α 0 ) = 0. Moreover, which implies that the selection of α in (A3) is possible.
The modified energy functional associated with problem (1)-(2) is given by where Direct differentiation of (23) using (1)-(2) leads to Lemma 2.1 ([32, 33] (Logarithmic Sobolev inequality)) Let u be any function in H 1 0 (Ω), and let a be any positive real number. Then Corollary 2.1 Let u be any function in H 2 0 (Ω), and let a be any positive real number. Then Proof Let f (s) = s ε 0 (| ln s|s). Note that f is continuous on (0, ∞), its limit at 0 + is 0 + , and its limit at ∞ is -∞. Then f has a maximum d ε 0 on (0, ∞), so (27) holds.

Existence results
In this subsection, we state without proof a local existence result of our problem (1)-(2).
The proof of Theorem 2.1 can be obtained by following the same arguments as in [8] and adapting the finite history to the infinite case. For the global existence, we have the following: where c 3 * is a positive embedding constant. Then we have: and The proof of Theorem 2.2 can be obtained by following the same arguments as in [8] by adapting the finite memory to infinite memory.

Technical lemmas
In this section, we start by establishing several lemmas needed for the proof of our main result.

Lemma 3.1 There exists a positive constant M 1 such that
where h 1 (t) := +∞ 0 Proof The proof is based on some arguments in [30]. In fact, we have where M 1 = max{2, 4E(0) }.

Lemma 3.2 Assume that h satisfies (A1).
Then, for u ∈ H 2 0 (Ω), Proof The proof can be easily obtained by applying the Cauchy-Schwarz and Poincaré inequalities.

Remark 3.1 Recalling
which gives Using (47), for any ε 0 ∈ (0, 1), we obtain that where q 0 > 0 is small enough, H is defined in Remark 2.2, and Proof To establish (49), we introduce the functional Then since E is nonincreasing, by (23) we get Thus q 0 can be chosen so small so that, for all t > 0, Without loss of generality, for all t > 0, we assume that λ(t) > 0; otherwise, we get an exponential decay from (39). Using Jensen's inequality, (2.3), (50), and (53) gives and hence (49) is established.

Decay result
In this section, we state and prove our main result and provide an example to illustrate our decay results. Let us start introducing some functions and then establishing several lemmas needed for the proof of our main result. As in [30], we introduce the following functions: where G -1 (t) = (H -1 (t)) 1 1+ε 0 and ε 0 ∈ (0, 1). Further, we introduce the class S of functions χ : R + → R * + satisfying, for fixed c 1 , c 2 > 0 (should be selected carefully in (76)), and where d > 0, c is a generic positive constant that may change from line to line, h 2 and q will be defined later in the proof of our main result, and Remark 4.1 According to the properties of H introduced in (A2) and the definition of G, we can see that G > 0 and G > 0 on (0, r], G 2 is convex increasing and defines a bijection from R + to R + , G 1 is decreasing and defines a bijection from (0, 1] to R + , and G 3 and G 4 are convex increasing functions on (0, r]. Then the set S is not empty because it contains χ(s) = εG 5 (s) with 0 < ε ≤ 1 small enough. Indeed, (57) is satisfied (since (55) and (59)). (29) hold. Then for any χ satisfying (57) and (58) and for any ε 0 ∈ (0, 1), there exists a strictly positive constant C such that the solution of (1)-(2) satisfies, for all t ≥ 0,

Theorem 4.1 Assume that (A1)-(A3) and
where G 5 and χ are defined in (55) and (57), respectively, and q will be defined later in the proof.

Conclusion
As far as we know, there are no decay results in the literature known for logarithmic plate equation with infinite memory and a wider class of relaxation functions. Our work extends the works for some wave equations treated in the literature to the plate equation with logarithmic nonlinearity. Also, we succeed to extend some general decay results, known for the case of finite history, to the case of infinite history, where the relaxation function satisfies a wider class of relaxation functions. Furthermore, we dropped the boundedness assumption on the history data considered in earlier results in the literature.