General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

On one hand, the study of elliptic, parabolic, and hyperbolic problems with nonlinearities of variable exponent type has been attracting much interest [1,2,19,21,31]. The nonlinearities of such type describe various physical applications, for example, electrorheological fluids [29], nonlinear elastics [33], non-Newtonian fluids [3], and image precessing [1]. In recent years, some authors studied the following wave equation with such nonlinearities: (1.11) In case k = 0 in (1.11), Messaoudi et al. [21] proved the local existence of solution and showed a blow-up result of the solution with negative initial energy when the exponents m(x) ≥ 2 and p(x) ≥ 2 satisfy some hypotheses. Later, in [12], the authors proved the global existence of solution for the same equation by giving some conditions on initial data. Moreover, they showed that the solution decays exponentially when m(x) = 2 and polynomially when m(x) ≥ 2 and m 2 > 2, where m 2 = ess sup x∈ m(x), by using an integral inequality introduced by Komornik [17]. In case k = 0 in (1.11), Park and Kang [27] obtained similar results of [21] for the solution with certain positive initial energy. Most recent, Messaoudi et al. [20] established very general decay results when m(x) > 1 and the relaxation function k satisfies (1.10). Their results generalize and extend the previous results for problem (1.11). Inspired by these works, in this article, we consider the viscoelastic von Karman system (1.1)-(1.5) with damping, source, and time delay effects of variable-exponent type. Time delay appears in the phenomena depending on some past occurrences as well as on the present state, and may cause instability. We refer to [7,32] for more applications of time delay and [11,16,24] for various decay results of delayed equations. In the absence of memory and time delay(k = β = 0) in (1.1), Ha and Park [13] proved the global existence of solution and showed exponential or polynomial decay results depending on m 2 ≥ 2. At this point, it is worth to say that there are no works on the global existence of solution and general decay of the solution for viscoelastic von Karman equations with variable exponent damping and source terms. Due to the presence of source effect, we have some difficulty in deriving desired general decay results. We overcome this by giving some conditions on initial data. Moreover, as far as we know, the global existence and decay of solutions for viscoelastic von Karman equations with delay of variable exponent type have not been considered before. Thus, we intend to discuss the issues for problem (1.1)-(1.5).
Here are the contents of this paper. We give preliminaries in Sect. 2. We show a global existence result in Sect. 3. We establish general decay results for both cases 1 < m 1 < 2 and m 1 > 2, where m 1 = ess inf x∈ m(x).

Preliminaries
In this section, we present notations, review necessary materials, give assumptions, and state a local existence result.
We denote by · Y the norm of a norm space Y . To simplify notations, we denote · L s ( ) as · s for 1 ≤ s ≤ ∞. We use the letter B s to denote the embedding constant Here, we recall Lebesgue and Sobolev spaces of variable exponents (see e.g. [8,9,18]). Let D be a bounded domain of R n , n ≥ 1, and r : → [1, ∞] be a measurable function. The Lebesque space It is said that r(·) satisfies the log-Hölder continuity condition if for all x,x ∈ D with |x -x| < b 2 , where b 1 > 0 and 0 < b 2 < 1. Throughout this paper, we let We remind the following property of von Karman bracket.
If at least one of them is an element of H 2 0 ( ), then We give the following assumptions.

Global existence
In this section, we derive the global existence of solution to problem (2.7)-(2.12). In the proof of global existence and decay results, we will use the following lemma several times.
Remark 3.1 Let the conditions of Lemma 3.3 hold. From the result of Lemma 3.3 and the same argument of (3.15), we also find From Lemma 3.2 and Lemma 3.3, we have the global existence result. Proof It suffices to show that u t From this and (3.17), we have

General decay results
In this section, we derive general decay results for both cases when m 1 ≥ 2 and when 1 < m 1 < 2 by following the ideas in [20] and [23] with some necessary modification. First, we let then, by the arguments of [14,23], we have the following lemma.

Lemma 4.3 The function ϒ satisfies
Proof Using (2.9) and y(x, 0, t) = u t (x, t), we get Proof Noting g (t) = -k(t) and using Young's inequality, we see From here, c and C i denote generic constants, c δ > 0 denotes a generic constant depending on δ > 0, and c δ (x) = m(x)-1 . We note that c δ (x) is bounded on for fixed δ > 0, that is, |c δ (x)| ≤ c δ for all x ∈ .

General decay for the case m 1 ≥ 2
In this subsection, we derive a general decay result for the case m 1 ≥ 2.