Study of solutions to a class of certain parabolic systems with variable exponents

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: ai x ð Þuit div rui j ji x ð Þ rui 1⁄4 fi x; u1; u2 ð Þ; i 1⁄4 1; 2 ð Þ: 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 blow-up results is shown, by using the energy method.


Introduction
Let X & R N (N ! 1) be a bounded Lipshitz domain and 0 < T < 1.It will be assumed throughout this paper that p(x) is continuous function defined in X with logarithmic module of continuity: for any x; y 2 X with x À y j j< We set Q T = X • (0, T) and R T = oX • (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: where p i ðxÞ 2 CðXÞ is a function, (i = 1, 2).
The operator ÀÁ pðxÞ w ¼ Àdiv rw j j pðxÞÀ2 rw is called p(x)-Laplacian, which will be reduced to the p-Laplacian 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 For the case p i (x) = p i > 2, and a i (x) = 1, (i = 1, 2), system (2) models as non-Newtonian fluids [2,21] and nonlinear filtration [2], etc.In the non-Newtonian 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(AE)-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 Àdivðjruj pðxÞÀ2 ruÞ 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][5][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 non-existence 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 equation where k > 0, m + k À 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 blow-up 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 blow-up results will be proved.The asymptotic behaviour of solution is established in Section 5.

Preliminaries
To consider problems with variable exponents, one needs the basic theory of spaces L p(x) (X) and W 1p(x) (X).For the convenience of readers, let us review them briefly here.The details and more properties of variable-exponent Lebesgue-Sobolev spaces can be found in [16,17].
The conjugate space is L q(x) (X), with As in the case of a constant exponent, set endowed with the norm ju j jj 1;pðxÞ ¼ ju j jj pðxÞ þ jru j jj pðxÞ : Similarly we also denote by W

Proposition 2.3 [16]
For u 2 W In the study of the global existence of solutions, we need the following hypotheses (H): 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, X, 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.

Existence
We will semi-discrete (2) in time and solve the corresponding elliptic problem.Based on the semi-discrete 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) where Ns = T and T is a fixed positive real, and 1 n N. ðXÞ; we consider the functional where g 2 L 1 ðXÞ is a known function.Using Young's inequality and Proposition 2.1, there exist constants C 1 , C 2 > 0, such that: hence U(v) ! 1, as jv j jj 1;p i ðxÞ !þ1: Since the norm is lower semi-continuous and R X gvdx is continuous functional, U(v) is weakly lower semi-continuous on W 1;p i ðxÞ 0 ðXÞ and satisfy the coercive condition.From [14] we conclude that there exists v Ã 2 W 1;p i ðxÞ 0 ðXÞ; such that: and v* is the weak solutions of the Euler equation corresponding to U(v), we obtain a weak solution u 1 i of (4).
M, we may prove by induction that (4) has a solution u n i in L 1 ðXÞ: We put u 1 i :¼ w i and for any integer k > 0, we may take ðw i À MsÞ k þ as a test function in (5) to get By using the Hölder's inequality and are well defined and satisfied in addition We first establish some energy estimates of u is ; ũis .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 Proof.(a) By lemma 3.3, for any n 2 N, u n i is bounded; whence (7) (b) Multiplying (4) by su n i , summing from n = 1 to N and integrating over X, we obtain By using the Young's inequality, for > 0 small, there exists With the aid of the identity 2a(a With the above estimates, we get (8).(c) Multiplying the equation ( 4) by u n i À u nÀ1 i and summing from n = 1 to N, we get s By using the Young's inequality, we get From the convexity of the expression R X rw j j p i ðxÞ dx; we get the following inequality: which imply with ( 12) and ( 13) that s 2 By lemma 3.4, there exists M i > 0 independent of s such that: Therefore, taking s !0 þ ; and up to subsequence, we get that there exists From ( 14), it follows that u i = v i .From (15), we get that By Aubin-Simon's compactness results [22], we have Now, multiplying (4) by u is À u i and using ( 14) and ( 15), we get by straightforward calculations: where o s ð1Þ !0 as s !0 þ .Thus, we get that and from ( 16) we have thus, and consequently by the same as that in [24] Á p i ðxÞ u is !Á p i ðxÞ u i in L p i ðxÞ 0 ð0; T ; W À1;p i ðxÞ 0 ðXÞÞ: Therefore, u i satisfies (3).
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 X, the difference we observe that w !ÀD p(x) w is monotone from W 1;pðxÞ 0 ðXÞ to W À1;pðxÞ 0 ðXÞ 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 ðu 1 ; u 2 Þ !ðu 1 ðtÞ; u 2 ðtÞÞ can be obtained similarly.
Remark 3.5 From Theorem 3.2, the solution of system (2) generates a semigroup SðtÞ Remark 3.6 If we assume that f i (x, s 1 , s 2 ) = g i (x, s 1 , s 2 ) À h i (x, s i ) where h i : Â R !R is Carathéodory mapping and that there exist positive constants L j , c j , m j , C j such that: and a; b 2 R; where a i ðxÞ 2 CðXÞ; with 2 By the same argument as that in [12,32], One can show in the same way that the semigroup S t ð Þ f g t!0 associated with system (2) admits an absorbing set in Where T 0 depends only on B.

Blow-up of solutions
In this section, we shall investigate the blow-up properties of solutions to system (2) using energy methods.To this end, we consider the following hypotheses on the data. (H4) Multiplying the first equation of ( 2) by u 1 , the second by u 2 , integrating by parts, we have By using Hölder's inequality, we have 1 2 where c 0 ¼ max ja 1 j jj 1 ; ja 2 j jj 1 À Á : we have by combining ( 23), ( 24) where A direct integration of the above inequality over (0, t) then yields g which implies that g(t) bows up at a finite time ; and so does u i .

Asymptotic behaviour
This section is devoted to the asymptotic behaviour of solutions.In order to prove the asymptotic behaviour, we assume Multiplying the second equation in (2) by u 2 and integrating over Summing up ( 25) and ( 26), we have from hypothesis (H5) that By u i 2 W 1;p i ðxÞ 0 ðXÞ; using Poincaré's inequality, we obtain According to the assumption that p 1 (x) p 2 (x), Then 2 < p À 2 : Hence, we get from (28) that 1 2 0; a:e; t !0: 0; or h 0 ðtÞ þ ChðtÞ The proof is complete.