Existence of Positive Solutions and Its Asymptotic Behavior of (p(x),q(x))-Laplacian Parabolic System

Hamza Medekhel 1,2, Salah Boulaaras 3,4,* , Khaled Zennir 3 and Ali Allahem 5 1 Department of Mathematics and Computer Science, Larbi Tebessi University, Tebessa 12002, Algeria; hamza-medekhel@univ-eloued.dz 2 Department of Mathematics, Exact Sciences Faculty, University of EL Oued, P.O. Box 789, El-Oued 39000, Algeria 3 Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Buraydah 51452, Saudi Arabia; k.zennir@qu.edu.sa 4 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran 31000, Algeria 5 Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia; aallahem@qu.edu.sa * Correspondence: saleh_boulaares@yahoo.fr or S.Boularas@qu.edu.sa


Introduction
In this paper, we consider the following evolutionary problem: find u ∈ L 2 (0, T, H 1 0 (Ω)) solution of where Ω ⊂ R N is a bounded domain and the functions p(x), q(x) belong to C 1 (Ω) and satisfying the following conditions: and satisfy some natural growth condition at u = ∞.
Symmetry 2019, 11, 332; doi:10.3390/sym11030332www.mdpi.com/journal/symmetry∆ p(x) is given by ∆ p(x) u = div(|∇u| p(x)−2 ∇u) is called p(x)-Laplacian, the parameters λ, λ 1 , λ 2 , µ 1 and µ 2 are positive with a, b, c, d are regular functions.In addition we did not consider any sign condition on f (0) , g (0) , h (0) , τ (0) .The linear and nonlinear stationary equations with operators of quasilinear homogeneous type as p-Laplace operator can be carried out according to the standard Sobolev spaces theory of W m,p , and thus we can find the weak solutions.The last spaces consist of functions having weak derivatives which verify some conditions of integrability.Thus, we can have the nonhomogeneous case of p(.)-Laplace operators in this last condition.We will use Sobolev spaces of the exponential variable in our standard framework, so that L p (.) (Ω) will be used instead of Lebesgue spaces L p (Ω) .
Here, in our study we consider the boundedness condition in domain Ω, because many results under p-Laplacian theory are not usually verified for the p(x)-Laplacian theory; for that in ( [6]) the quotient becomes 0 generally.Then λ p(x) can be positive only for some given conditions.In fact, the first eigenvalue of p(x)-Laplacian and its associated eigenfunction cannot exist, the existence of the positive first eigenvalue λ p and getting its eigenfunction are very important in the p-Laplacian problem study.Therefore, the study of existence of solutions of our problems have more meaning.Many studies of the experimental side have been studied on various materials that rely on this advanced theory, as they are important in electrical fluids, which states that viscosity relates to the electric field in a certain liquid.Recently, in ([2,6-8]), we have proved the existence of positive solutions of many classes of (p(x), q(x))-Laplacian stationary problems by using the sub -super solution concept.The current results are an extension of our previous stationary study to the parabolic case, where we follow-up the same procedures mathematical proofs similar to that in ( [2,7]) by using difference time scheme taking into consideration the stability analysis of the used scheme and the same conditions which have given in references mentioned earlier.Our result is an extension for our previous study in ( [2,7,9]) which studied the stationary case, this idea is new for evolutionary case of this kind of problem.
The outline of paper consists as follow: In first section we give some definitions, basic theorems and necessarily propositions in the functional analysis which will be used in our study.Then in Section 3, we prove our main result.

Preliminaries Results and Assumptions
In order to discuss problem (1), we need some theories on W 1,p(x) 0 (Ω) which we call variable exponent Sobolev space.Firstly we state some basic properties of spaces W 1,p(x) 0 (Ω) which will be used later (for details, see [10]).

The Semi-Discrete Problem
We discrete the problem (1) by difference time scheme, we obtain the following problems where Nτ = T, 0 < τ < 1, and for 1 ≤ k ≤ N.
According to ([11] in Theorem 3.1), the bounded operator L : W (Ω) * is a continuous and strictly monotone, and it is a homeomorphism.We considere mapping A : W where h(x, u k ) is continuous on Ω × R, and h(x, .) is increasing function.It is easy to verify that A is a continuous bounded mapping.By the proof ( [12]).
(2) We say called a sub solution (respectively a super solution) of (1) if Here, we will use the notation d(x, ∂Ω) to denote the distance of x ∈ Ω to denote the distance of Ω. Denote and Lemma 2. ( [13]), If positive parameter η is large enough and w is the unique solution of ( 7), then we have (i) For any θ ∈ (0, 1) there exists a positive constant C 1 , such that (ii) There exists a positive constant C 2 , such that

Main Result
In the following, once we have no misunderstanding, we always use C i to denote the positive constants.
Theorem 1. Assume that the conditions (H 1 )-(H 5 ) are statisfied.Then, problem (1) has a positive solution when λ is large enough.
Step 1.We will construct a subsolution of (1).Let σ ∈ (0, δ) is small enough.Denote and It easy to see that and By some simple computations we obtain From (H 3 ) there exists a positive constant M > 1 such that If k is sufficiently large, from ( 8), we have Let λξ = kα, then k p(x) α −λ p(x) ξ.
Step 2. We will construct a supersolution of problem (1), we consider where We shall prove that (z k 1 , z k 2 ) is a supersolution of problem (1).From Lemma 2, we have For ψ ∈ W 1,q(x) 0 (Ω) with ψ 0, it is easy to see that By (H 4 ), for µ a large enough, using Lemma 2, we have Hence Also, for ϕ ∈ W 1,p(x) (Ω) with ϕ ≥ 0, it is easy to see that By (H 3 ), (H 4 ) and Lemma 2, when µ is sufficiently large, we have According to ( 16) and (17), we can conclude that (z k 1 , z k 2 ) is a supersolution of problem (1).It only remains to prove that φ k In the definition of v 1 (x), let We claim that From the definition of v 1 , it is easy to see that From the definition of v 1 ,we have e Ω, where θ ∈ (0, 1).
Under the comparaison principle, we have From ( 18) and (19), when η λ p+ and λ 1 is sufficiently large, we have According to the comparaison principle, when µ is large enough, we have Combining the definition of v 1 (x) and (20), it is easy to see that When µ 1 and λ is a large enough, from Lemma 2, we can note that is a large enough.Similarly, we have φ k 2 (x) ≤ z k 2 (x).This completes the proof.

Asymptotic Behavior of Solutions
Definition 2. A measurable funtion u : Ω T → R is an weak solution to hyperbolic systems involving of (p(x), q(x))−Laplacien (1) in Lemma 4. Let u, u be the solutions of (1) with u (x, 0) = ϕ 1 , u (x, 0) = ϕ 2 Than u(x, t) is nondercreasing in t , u (x, t) is nonincreasing and u > u for all t ≥ 0, x ∈ Ω Theorem 2. Let hypotheses (H 1 ), (H 2 ) and (H 3 ) be satisfied.and let u (x, t) the solution of a new class of hyperbolic systems (1) with Ψ ∈ S * than lim t→∞ u (x, t) = u s (x) if u s ≤ Ψ ≤ u s u s (x) if u s ≤ Ψ ≤ u s .
Proof.The pair (u s , u s ) and the pair ( u s , u s ) are both sub-super solutions of (4), the maximale and minimale property of u s and u s in S * ensures that: u s is the unique solution in [ u s , u s ] and u s is the unique solution in [u s , u s ].

Conclusions
Our result is an extension for our previous study in ( [2,7,8]) which studied the stationary case, this idea is new for evolutionary case of this kind of problem, This paper deals with the existence of positively solution and its asymptotic behavior for parabolic system of (p(x), q(x))-Laplacian system of partial differential equations using a sub and super solution according to some given boundary conditions, which is familiar in physics, since it appears clearly natural in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics (see [3,14]).This sort of problem has many applications in several branches of physics such as nuclear physics, optics, and geophysics (see [7,15]).In future work, we will try to extend this study for the hyperbolic case of the presented problem, but by using the semigroup theory.
Thus, (18) is valid.Obviously, there exists a positive constants C 3 , such that γ ≤ C 3 λ.Since d(x) ∈ C 2 (∂Ω 3δ ), according to the proof of Lemma 2, there exists a positive constant C 4 , such that