Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy

Abstract: In the present paper, we consider an important problem from the point of view of application in sciences and mechanic, namely, a class of p(x)-Laplacian type parabolic equation with weakviscoelasticity. Here, we are concerned with global in time non-existence under suitable conditions on the exponents q(x) and p(x) with positive initial energy. We show that the weak-memory term is unable to stabilize problem (1.2) under conditions (1.5) and (1.7). Our main interest in this paper arose in the first place in consequence of a query to blow-up phenomenon.

describes the evolution of the concentration during the propagation of particles in Ω. Here c(x, t, u) describes a source if it is positive or a bowl if it is negative, the diffusion coefficient a(x, t, u, ∇u) reflects the intrinsic ability of diffusion in particles in the medium. Needful to say, it has a numerous generalizations, we can also do with p(x)-Laplacian denoted by ∆ p(x) , which has an exponent variable property.
The fact that p(x)-Laplacian is not homogeneous, makes non-linearities more complicated than the operator p-Laplacian. Studies of various mathematical systems with variable exponent growth conditions have received considerable attention in recent years, which is justified by their various physical applications. However, few papers have treated evolutionary equations of non-local p(x)-Laplacian type (Please see [1,5,12,13,24,25]). Viscoelastic materials demonstrate properties between those of elastic materials and viscous fluid. As a consequence of the widespread use of polymers and other modern materials which exhibit stress relaxation, the theory of visco-elasticity has provided important applications in materials science and engineering (Please see [7,8,10,11,19,21,22]).
The viscoelastic materials show a behavior which is something between that of elastic solids and Newtonian fluids. Indeed, the stresses in these media depend on the entire history of their deformation, not only on their current state of deformation or their current state of motion. This is the reason why they are called materials with memory. The viscoelastic equations with fading memory in a bounded space has been deeply studied by several authors, in view of its wide applicability.
The lack of stability of solutions of viscoelastic partial differential equations is a huge restriction for qualitative studies. In the present paper, we consider for x ∈ Ω, 0 < t < ∞ with initial and boundary conditions where q(·) and p(·) are two continuous functions on Ω such that The viscoelastic term is represented as t 0 µ (t − s) ∆ x w ds, it is called "weak-viscoelastic" when it comes with the time weighted function σ(t), which is considered as a dissipative term and causes stability of systems. The nonlinear term |w| p(x)−2 w is known as the source of instability. The importance of our study lies in the study of the interaction between the exponents of source term and the Laplacian with the presence of weak-viscoelastic term. We take the exponents as a variable functions, with their difficulties in the mathematical point of view, to obtain a very large applications. These contributions extend the early results in literature.
We assume that q(x) satisfies the Zhikov-Fan condition, i.e. for all x, y ∈ Ω, with K > 0, 0 < κ < 1 and Since the relaxation function µ lies with the stability of solutions, we state assumptions on µ and σ as: µ, σ ∈ C 1 (R + , R + ) satisfy For positive constant C depending only on Ω determined by Lemma 2.1, we set for some constant λ > 0 (will be specified later) is related to the parabolic problem and when the exponents q(x) = q, p(x) = p, the existence/non-existence results have been extensively studied (please see [2-4, 14, 15, 23]).
Extinction phenomenon for parabolic equation with nonlinearities in divergence form are investigated in [18], under nonlinear boundary flux in bounded star-shaped region. The authors assumed conditions on weight function to guarantee that the solution exists globally or blows up at finite time. Moreover, using the modified differential inequality, upper and lower bounds for the blow-up time of solutions were derived in higher dimensional spaces.
In the case where q(·) and p(·) are constants, the existence of local solutions of initial-boundary value problem is proved by Akagi in [2] for initial data u 0 ∈ L r (Ω) and 2 ≤ q < p < +∞ and Ω is an open bounded domain in R n , under r > n(p − q)/q. For the case where q(·) and p(·) are two measurable functions, it is well known that some additional techniques must be used to study the existence/nonexistence of solutions for (1.2)-(1.4) and of course the classical methods may be failed unless some developments are made. For the case of nonlocal p(x)-Laplacian equations and in the absence of the memory term (µ ≡ 0), problem (1.2)-(1.4) has been studied byÔtani [20]. The author treated the question of existence and qualitative studies of solutions of (1.2)-(1.4) and showed that the difficulties come from the use of non-monotone perturbation theory. To complete these studies, the questions of blow-up of solutions for the same problem are discussed later by many authors.
In this paper we shall establish a blow-up result of solutions for problem (1.2)-(1.4) in the Lebesgue and Sobolev spaces with variable exponents and positive initial energy.

Preliminary and well known results
We try to list here some useful mathematical tools. First, let p : Ω −→ (1, ∞) be a measurable function. Denoting by We define the p(·) modular of a measurable function w : where The special Orlicz Musielak space L p(·) (Ω) is a Lebesgue space with variable-exponent and it consists of all the measurable function w defined on Ω for which be the Luxembourg norm on this space (see [16]). The Sobolev space W 1,q(·) (Ω) consists of functions w ∈ L q(·) (Ω) whose distributional gradient ∇ x w exists and satisfies |∇ x w| ∈ L q(·) (Ω). This space is a Banach with respect to the norm w 1,q(·) = w q(·) + ∇ x w q(·) .
where Ω is a bounded domain and C > 0 is a constant. The norm of space W 1,q(·) 0 (Ω) is given by is a measurable function and with continuous and compact embedding and where C S > 0 is an embedding constant.

Main results
Here, we present without proof, the first known result concerning local existence (in time) for problem (1.2)-(1.4). (see [22]) Now, to prove the blow up result, we should define the energy functional E (t), associated with our problem by Proof. Multiplying (1.2) by ∂ t w, integrating by parts over Ω, using (1.7) and Lemma 2.6, we get the desired result.
By using conditions (1.6), (1.8) and thanks to there exists a constant by (1.5) and (1.7), we have 1 such that Let f be a function defined by f : Then f is increasing in (0, α), decreasing for ψ > α, f (ψ) −→ −∞ as ψ −→ +∞ and Then there exists a constant β > α such that
Combining with the estimates obtained in the above Lemmas, we state the main result concerned with finite time blow-up.
By differentiating L, we get Using Cauchy Schwarz's inequality and Lemma 2.6 to obtain for some positive constant C 0 > 0 (to be determined later). Then, we have By using Young's inequality, (1.8) and Lemma 2.6 for some constant c > 0, we obtain We then substitute for σ(t) (µ • ∇ x w) (t) from (3.2), hence (3.8) becomes . By using (1.8) and (3.6), the estimate (3.9) takes the form This implies that, we can choose c > C 0 + λ to get Then