Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

: In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: where γ ( . ) is a function satisfying some conditions. We ﬁrst prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some speciﬁc conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature.


Introduction
In this paper we are concerned with the following problem b(t − s)∆u(s)ds + |u t | γ(·)−2 u t = u ln |u| α in Ω × (0, +∞), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), in Ω, where Ω is a bounded domain of R n with a smooth boundary ∂Ω, ν is the unit outer normal to ∂Ω, u 0 and u 1 are the given data, b is a relaxation function and γ(.) is a variable exponent.
Problem (1.1) contains three class of problems: I. Viscoelasticity with wide class of relaxation functions. The importance of the viscoelastic properties of materials has been realized because of the rapid developments in rubber and plastics 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 [1]. There is an extensive literature on the stabilization of viscoelastic wave equations and many results have been established. There are a lot of contributions to generalize the decay rates by allowing an extended class of relaxation functions and give general decay rates. In fact, the journey of generalization of relaxation functions passed through several steps, we mention here the following stages: 1) As in [2], the relaxation function b satisfies , for two positive constants a 1 and a 2 , −a 1 b(t) ≤ b (t) ≤ −a 2 b(t), t ≥ 0.
With the advancement of sciences and technology, many physical and engineering models required more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, the elecrtorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through a porous media and image processing. More details on these problems can be found in [8,9]. For hyperbolic problems involving variable-exponent nonlinearities, we refer to [10][11][12][13][14][15]. For more results of other problems with the nonlinearity of power type, we refer the interested reader to see [16][17][18]. III. Logarithmic source term.
The logarithmic nonlinearity appears naturally in inflation cosmology and supersymmetric filed theories, quantum mechanics and nuclear physics [19,20]. Problems with logarithmic nonlinearity have a lot of applications in many branches of physics such as nuclear physics, optics and geophysics [21][22][23].
In this paper, we consider problem (1.1) and prove the global existence of solutions, using the well-depth method. We then establish explicit and general decay results of the solution under suitable assumptions on the variable exponent γ(.) and very general assumption on the relaxation function. To the best of our knowledge, such a problem has not been discussed before in the context of nonlinearity with variable exponents.
In addition to the introduction, this paper has four other sections. In Section 2, we present some preliminaries. The Existence is given in Section 3. In Section 4, we establish some technical lemmas needed for the proof of the main results. Our stability results and their proof are given in Section 5.

Preliminaries
In this section, we present some preliminaries about the logarithmic nonlinerity and the Lebesgue and Sobolev spaces with variable exponents (see [24][25][26][27]). Throughout this paper, c is used to denote a generic positive constant.

Existence
In this section, we state the local existence theorem whose proof can be established by combining the arguments of [10,30,31]. Also, we state and prove a global existence result under smallness conditions on the initial data (u 0 , u 1 ).
We define the following functionals which are needed for establishing the global existence where for v ∈ L 2 loc (R + ; L 2 (Ω)), E(t) represents the modified energy functional associated to problem (1.1).
which shows that the soultion is global and bounded in time.

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

(4.21)
Therefore, for almost every x ∈ Ω, we have By using Young's, Hölder's and Poincaré's inequalities and Lemma 4.2, we get where Similarly, we have Therefore, For the last term in (4.17), the use of (2.4), Young's, Cauchy-Schwarz' and Poincaré's inequalities, the embedding theorem and Lemma 4.2 leads to, for any δ > 0, Combining the above estimates with (4.17), we obtain (4.15). For the proof of (4.16), we re-estimate the fifth term in (4.17) as follows:  and Proof. Since b is positive and b(0) > 0 then, for any t 0 > 0, we have By using (4.1), (4.2) and (4.15), then, for t ≥ t 0 and any λ 0 > 0, we have Using the Logarithmic Sobolev inequality, for 0 < λ 0 < 1 2 , we get At this point, we select λ 0 and α so small that Then, we choose N 2 large enough so that: and then N 1 large enough that Therefore, we arrive at the desired result (4.27). On the other hand, we can choose N 1 even larger (if needed) so that L ∼ E. (4.29)

Decay results
In this section, we establish our main decay results. For this purpose, we need the following remarks and lemma.

Proof. Case 1: B is linear
We multiply (4.27) by a(t) and use (5.1) and (5.2) to get Multiply (5.11) by a (t)E (t), and recall that a ≤ 0, to obtain Use of Young's inequality, with q = + 1 and q * = +1 , gives, for any ε > 0, We then choose 0 < ε < λ 0 c and use that a ≤ 0 and E ≤ 0, to get, for where L 1 = a +1 E L + cE. Then L 1 ∼ E (thanks to (4.29)) and Integrating over (t 0 , t) and using the fact that L 1 ∼ E, we obtain (5.9).
Case 2: B is non-linear. Using (4.27), (5.1) and (5.3), we obtain, ∀t ≥ t 0 , Combining the strictly increasing property of B and the fact that 1 t < 1 whenever t > 1, we obtain then, (5.12) becomes, for ∀t ≥ t 1 = max {t 0 , 1}, . (5.14) Set Using the facts that B > 0 and B > 0 on (o, r], (5.14) reduces to Now, for ε 1 < r and using (5.36) and the fact that E ≤ 0, B > 0, B > 0 on (0, r], we find that the functional L 2 , defined by satisfies, for some c 1 , c 2 > 0. and, for all t ≥ t 1 , · E(t) E(0) and α 2 = B −1 (χ(t)), we arrive at Then, multiplying (5.19) by a(t) and using (5.4), (5.15), we get Using the non-increasing property of a, we obtain, for all t ≥ t 1 , Therefore, by setting L 3 := aL 2 + cE ∼ E, we conclude that This gives, for a suitable choice of ε 1 , An integration of (5.20) yields Using the facts that B , B > 0 and the non-increasing property of E, we deduce that the map t → · E(t) E(0) is non-increasing and consequently, we have , ∀t ≥ t 1 (5.23) Next, we set B 2 (s) = sB (ε 1 s) which is strictly increasing, and consequently we obtain, , Finally, we infer This finishes the proof.
The following examples illustrate the results of Theorem 5.6: where a is a fixed constant. Then, (5.33) gives, for t > t 1 and ∈ (0, 1), . (5.48) Remark 5.7. The classical power-type nonlinearity term in [33] provides a canonical description for the dynamics analysis of a quasi-wave propagation in a nonlinear process, therefore, the fast cumulative of such nonlinear interactions results in a significant effect to the solution under large spatial and temporal scales. However, the logarithmic nonlinearity in (1.1) only expresses slowly cumulative of nonlinear, thus giving another kind of description for dynamic process. Let us note here that though the logarithmic nonlinearity is somehow weaker than the polynomial nonlinearity, both the existence and stability result are not obtained by straightforward application of the method used for polynomial nonlinearity.