Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 8, 2017



Article Number  A11  
Number of page(s)  8  
DOI  https://doi.org/10.1051/smdo/2017004  
Published online  18 August 2017 
Research Article
Study of solutions to a class of certain parabolic systems with variable exponents
^{1}
Research laboratory in Engineering (LRI), University Hassan II of Casablanca, National Higher School of Electricity and Mechanics (ENSEM), BP 8118, Oasis, Morocco
^{2}
Département MIG, Ecole Hassania des Travaux Publics (EHTP), BP 8106, Oasis, Casablanca, Morocco
^{*} email: h.elouardi@ensem.ac.ma
Received:
10
April
2017
Accepted:
22
June
2017
In this paper, the authors study an initial and boundary value problem to a system of evolution p(x)Laplacian systems coupled with general nonlinear terms:
a_{i}(x)u_{it} − div(∇u_{i}^{pi(x)−2}∇u_{i}) = f_{i}(x, u_{1}, u_{2}), (i = 1, 2).
The authors translate the parabolic equation into the elliptic equation by using the time discretization method, and then the existence and uniqueness solution are obtained. The blowup results is shown, by using the energy method.
Key words: p_{i}(x)Laplacian systems / Existence / Uniqueness / Variable exponent / BlowUp / Semidiscretization
© H. El Ouardi et al., published by EDP Sciences, 2017
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Let (N ≥ 1) be a bounded Lipshitz domain and 0 < T < ∞. It will be assumed throughout this paper that p(x) is continuous function defined in with logarithmic module of continuity: (1)
We set Q _{ T } = Ω × (0, T) and Σ_{ T } = ∂Ω × (0, T). Our aim is to prove the existence and uniqueness of solutions u = (u _{1}, u _{2}) to the nonlinear (p _{1}(x), p _{2}(x))Laplacian system:(2)where is a function, (i = 1, 2).
The operator is called p(x)Laplacian, which will be reduced to the pLaplacian when p(x) = p a constant.
The (p _{1}(x), p _{2}(x))Laplacian system (2) can be viewed as a generalization of (p, q)Laplacian system(3)
For the case p _{ i }(x) = p _{ i } > 2, and a _{ i }(x) = 1, (i = 1, 2), system (2) models as nonNewtonian fluids [2, 21] and nonlinear filtration [2], etc. In the nonNewtonian fluids theory, (p _{1}, p _{2}) is a characteristic quantity of the fluids, there have been many results about the existence, uniqueness of the solutions. We refer the readers to the bibliography given in [7, 9, 10, 11, 28, 31] and the references therein.
In recent years, the research of nonlinear problems with variable exponent growth conditions has been an interesting topic. p(·)growth problems can be regarded as a kind of nonstandard growth problems and these problems possess very complicated nonlinearities, for instance, the p(x)Laplacian operator is inhomogeneous. And these problems have many important applications in nonlinear elastic, electrorheological fluids and image restoration. The reader can find in [14, 22] several models in mathematical physics where this class of problem appears.
The case of a single equation of the type (2) has been studied in [4–6, 20] and the authors established the existence and uniqueness results, in [20], the authors use the difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p _{ i }(x).
The more interesting question concerning parabolic systems of (p _{1}(x), p _{2}(x))Laplacian type is to understand the asymptotic behavior of solutions when time goes to infinity. The study of the asymptotic behavior of the system is giving us relevant information about the structure of the phenomenon described in the model.
Concerning the elliptic systems with variable exponents, the results about existence and nonexistence are proved in [26, 27, 29, 30].
Note that system (2) has a more complicated nonlinearity than the classical (p,q)Laplacian system since it is nonhomogenous.
Recently, [24] study the equation the p(x)Laplacian equationwhere λ > 0, m + λ − 2 > 0 and a(x) is a positive continuous function. They examine under which conditions on behavior of a(x) corresponding nonnegative solutions of the Cauchy problems possess the finite speed of propagations or interface blowup phenomena.
In this paper, we consider the existence and uniqueness for the problem of the type (2) under some assumptions. The proof consists of two steps. First, we prove that the approximating problem admits a global solution; then we do some uniform estimates for these solutions. We mainly use skills of inequality estimation and the method of approximation solutions. By a standard limiting process, we obtain the existence to problem of the type (2).
The outline of this paper is the following: In Section 2, we introduce some basic Lebesgue and Sobolev spaces and state our main theorems. In Section 3, we give the existence and uniqueness of weak solutions. In Section 4, the blowup results will be proved. The asymptotic behaviour of solution is established in Section 5.
2 Preliminaries
To consider problems with variable exponents, one needs the basic theory of spaces L ^{ p(x)}(Ω) and W ^{1p(x)}(Ω). For the convenience of readers, let us review them briefly here. The details and more properties of variableexponent LebesgueSobolev spaces can be found in [16, 17].
Let When p ^{−} > 1, one can introduce the variableexponent Lebesgue spaceendowed with the Luxemburg norm.
The conjugate space is L ^{ q(x)}(Ω), with
As in the case of a constant exponent, setendowed with the norm
Similarly we also denote by the closure of in and is the dual of with respect to the inner product in
In Propositions 2.1–2.3, we describe some results about the Luxembourg norm.

The space is a separable, uniformly convex Banach space, and its conjugate space is where For any and we have the following Höldertype inequality:

If for any the imbedding is continuous, the norm of the imbedding does not exceed
Proposition 2.2 [16]

w_{(r(x))} < 1(= 1; > 1) ⇔ ρ(w) < 1 (= 1; > 1);


w_{(r(x))} → 0 ⇔ ρ(w) → 0; w_{(r(x))} → ∞ ⇔ ρ(w) → ∞.
Proposition 2.3 [16]
For there exists a constant such that:
This implies that and are equivalent norms of
System (2) does not admit classical solutions in general. So, we introduce weak solutions in the following sence.
Definition 2.4 A function u = (u _{1} , u _{2}) is said to be a weak solution of equation (2) , if the following conditions are satisfied:
In the study of the global existence of solutions, we need the following hypotheses (H):

(H1)

(H2)

(H3)
3 Main results
Remark 3.1 In this paper, we shall denote by c _{ i }, C _{ i } differents constants, depending on p _{ i }(x), T, Ω, but not on n, which may vary from line to line. Sometimes we shall refer to a constant depending on specific parameters C _{ i }(T), etc.
Our main existence result is the following:
Theorem 3.2 Let (H1)–(H3) hold. Then system (2) admits a unique solution u = (u _{ 1 } ,u _{ 2 }) ∈ (C((0, T); L ^{2}(Ω)))^{2} . Moreover, the mapping (φ _{1}, φ _{2}) → (u _{1}(t), u _{2}(t)) is continuous in L ^{2}(Ω) × L ^{2}(Ω).
Proof of the main results.
3.1 Existence
We will semidiscrete (2) in time and solve the corresponding elliptic problem. Based on the semidiscrete problem, we construct the corresponding approximate solutions. The key procedure is to establish necessary a priori estimates for finding the limit of the approximate solutions via a compactness argument.
We first consider the discrete scheme (4)(4)where Nτ = T and T is a fixed positive real, and 1 ≤ n ≤ N.
Lemma 3.3 For any fixed n, if Problem (4) admits a weak solution
Proof. On the space we consider the functionalwhere is a known function. Using Young’s inequality and Proposition 2.1, there exist constants C _{1}, C _{2} > 0, such that:hence Φ(v) → ∞, as Since the norm is lower semicontinuous and is continuous functional, Φ(v) is weakly lower semicontinuous on and satisfy the coercive condition. From [14] we conclude that there exists such that:and v* is the weak solutions of the Euler equation corresponding to Φ(v),
Choosing we obtain a weak solution of (4).(5)
Since , we may prove by induction that (4) has a solution in We put and for any integer k > 0, we may take as a test function in (5) to get
By using the Hölder inequality and , we have
We deduce
Letting k → ∞, we get Consider , we get easily that , i.e. and if we choose such that , we obtain
This completes the proof of lemma 3.3.
Now, we define the approximate solutions as set by: for all n ∈ {1, …, N}.are well defined and satisfied in addition(6)
We first establish some energy estimates of .
We need several lemmas to complete the proof of Theorem 3.2.
Lemma 3.4 There exists a positive constant C(T, u _{0}) such that, for all n = 1, …, N (7) (8) (9)and(10)
Proof. (a) By lemma 3.3, for any n ∈ N, is bounded; whence (7)
(b) Multiplying (4) by , summing from n = 1 to N and integrating over Ω, we obtain(11)
By using the Young inequality, for small, there exists such that(12)
With the aid of the identity 2α(α − β) = α ^{2} − β ^{2} + (α − β)^{2}, we get
With the above estimates, we get (8).
(c) Multiplying the equation (4) by and summing from n = 1 to N, we get
By using the Young inequality, we get(13)
From the convexity of the expression we get the following inequality:(14)
which imply with (12) and (13) that
By lemma 3.4, there exists M _{ i } > 0 independent of such that:(15)
Therefore, taking and up to subsequence, we get that there exists such that , and as ,(16) (17)
From (14), it follows that u _{ i } = v _{ i }. From (15), we get that(18)
By AubinSimon’s compactness results [22], we have(19)
Now, multiplying (4) by and using (14) and (15), we get by straightforward calculations:where as .
Thus, we get that(20)and from (16) we have thus,and consequently by the same as that in [24]
Therefore, u _{ i } satisfies (3).
3.2 Uniqueness
Let (H1)–(H3) be satisfied. Then system (2) has a unique solution u = (u _{1}, u _{2}) in Q _{ T. }
Proof. Let u = (u _{1}, u _{2}) and v = (v _{1}, v _{2}) be solutions of (2), we have:
Since f _{ i }(x,.,.) is locally Lipschitz uniformly in Ω, the difference w _{ i } = u _{ i } − v _{ i } satisfieswe observe that w → −Δ_{ p(x)} w is monotone from to (21)
We finally deduce from Gronwall’s lemma,
Thus, we deduce that u _{ i } = v _{ i }.
Thus the solution is unique. The continuity of the the mapping can be obtained similarly.
Remark 3.5 From Theorem 3.2, the solution of system (2) generates a semigroup in L ^{2}(Ω) × L ^{2}(Ω).
Remark 3.6 If we assume that f _{ i }(x, s _{1}, s _{2}) = g _{ i }(x, s _{1}, s _{2}) − h _{ i }(x, s _{ i }) where is Carathéodory mapping and that there exist positive constants L _{ j }, c _{ j }, m _{ j }, C _{ j } such that:

for any x ∈ Ω and

for any x ∈ Ω and where with and
By the same argument as that in [12, 32], One can show in the same way that the semigroup associated with system (2) admits an absorbing set in there is a bounded set such that, for any bounded set B in L ^{2}(Ω) × L ^{2}(Ω), there exists a T _{0} > 0 such that for any t ≥ T _{0}. Where T _{0} depends only on B.
4 Blowup of solutions
In this section, we shall investigate the blowup properties of solutions to system (2) using energy methods. To this end, we consider the following hypotheses on the data.
Throughout this section, we define for t ≥ 0
Theorem 4.1 Let (H1)–(H5) be satisfied, then the solutions of system (2) blow up in finite time, namely, there exists a T* < ∞ such that as t → T*.
Proof. We define .
Multiplying the first equation of (2) by , the second by , integrating by parts, we have(22)which implies that E(t) ≤ E(0).
Next define
Multiplying the first equation of (2) by u _{1}, the second by u _{2}, integrating by parts, we have(23)
By using Hölder’s inequality, we have(24)where
By the formula we have by combining (23), (24)where
A direct integration of the above inequality over (0, t) then yieldswhich implies that g(t) bows up at a finite time and so does u _{ i }.
5 Asymptotic behaviour
This section is devoted to the asymptotic behaviour of solutions. In order to prove the asymptotic behaviour, we assume
(H6)
Theorem 5.1 The weak solution u = (u _{1}(t), u _{2}(t)) obtained in Theorem 3.2, satifies: where C _{ i } > 0 (i = 1, 2, 3), or or .
Proof. Let u _{ i } be solution of (2).
Multiplying the first equation in (2) by u _{1} and integrating over Q _{ T },(25)
Multiplying the second equation in (2) by u _{2} and integrating over Q _{ T },(26)
Summing up (25) and (26), we have from hypothesis (H5) that(27)
By u _{ i } using Poincaré’s inequality, we obtain(28)
If and by Proposition 2.2,(29)
According to the assumption that p _{1}(x) ≤ p _{2}(x), Then
Hence, we get from (28) that(30)
By the formula we have(31)this implies that(32)where C _{3} = min(C _{1}, C _{2}).
Then, we obtain from (32) and (H2) that(33)
Then we get from (28) that(34)
Similarly, if and or and we can also obtain the similar results
The proof is complete.
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Cite this article as: El Ouardi H & Ghabbar Y: Study of solutions to a class of certain parabolic systems with variable exponents. Int. J. Simul. Multisci. Des. Optim., 2017, 8, A11.
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