Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

In this paper, we prove the existence of at least one periodic solution for some nonlinear parabolic boundary value problems associated with Leray–Lions’s operators with variable exponents under the hypothesis of existence of well-ordered sub- and supersolutions.


Introduction
Let be a bounded open set of R N (N ≥ 1) with a smooth boundary ∂ , and fixed T > 0. Our aim here is to prove the existence of periodic solutions for the following nonlinear parabolic problem (x, t, u, ∇u) in × (0, T ), where Au = −div(A(·, ·, u, ∇u)) is a Leray-Lions's type operator with variable exponents acting from some functional space V 0 into its topological dual V 0 and where f is a nonlinear Carathéodory function, whose growth with respect to |∇u| is at most of order p(x) in the sense defined below (Hypothesis A4). The suitable functional spaces to deal with in this type of problems are generalized Lebesgue and Sobolev spaces L p(x) ( ) and W 1, p(x) ( ), respectively. There are many differences between Lebesgue and Sobolev spaces with constant exponents and those with variable exponents. For instance, p(x) needs to satisfy the log-Höder condition (see [10,12]) in order that the Poincaré's inequality and the density of smooth functions in W 1, p(x) ( ) hold. Many difficulties arise in the case of variable exponents. One typical difficulty when dealing with problems like (P) is to define adequate functional spaces for solutions. When p(x) = p is a constant, it is well known that L p (0, T ; W 1, p 0 ( )) can be taken as a space of solutions. However, when p(x) is nonconstant, then nor L p(x) (0, T ; W 1, p(x) 0 ( )) neither L p − (0, T ; W 1, p(x) 0 ( )), where p − = min p(x), constitute a suitable space of solutions (see [5].) Henceforth, to overcome this difficulty, we shall define below our functional space of solutions V 0 as it was done by Bendahmane in [5].
Nonlinear problems defined by (P) arise in many applications; for instance, in electrorheological fluids (see [18]), where the essential part of the energy is given by |Du(x)| p(x) dx (Du being symmetric part of ∇u). This type of fluids has the ability to change its mechanical properties (for example becoming a solid gel) when an electric field is applied. Another important application is when f depends only on (x, t) and A(x, t, s, ξ) = |ξ | p(x)−2 ξ ; then the problem (P) can be seen as a sort of nonlinear diffusion equation whose coefficient of diffusion takes the form |∇u| p(x)−2 by analogy with Fick's diffusion model (see [2]). For other applications, we refer the reader to [7,21].
There is by now an extensive literature on the existence of solutions for problem (P). Let us recall some known results in the case where p(x) := p is a real constant. In [9], by applying a penalty method to an appropriately associated auxiliary parabolic variational inequality, J. Deuel and P. Hess proved the existence of at least one periodic solution for problem (P) in the case where the natural growth of f with respect to |∇u| is of order less than p, that means | f (x, t, u, ∇u)| ≤ k(x, t) + c|∇u| p−δ for some δ > 0 and k(x, t) ∈ L 1+δ ( × (0, T )), c being a positive constant. In [14], N. Grenon extends the result of [9] to the case where the natural growth of f with respect to |∇u| is at most of order p; but instead of a periodicity condition the author considered an initial one. The proof therein is based on some regularization techniques used in [6,17].
Let us point out that in the two previous works, the hypothesis of existence of well-ordered sub-and supersolutions is supposed. Following [9], the results in [14] were extended by El Hachimi and Lamrani in [11], where the authors obtained the existence of periodic solutions, under the same hypotheses as in [14]. For variable exponents, this kind of problems has been studied by many authors [2,5,13,20], by means of different methods such as: subdifferential operators, Galerkin scheme, semigroup theory, etc.
The main goal of this paper is to extend the results in [11] to the variable exponents case under the hypothesis of existence of well-ordered sub-and supersolutions. It is well known that this method, when it is applicable, has more advantages compared to other methods. For example, we can give some order properties of the solutions. Nevertheless, this method is quite complicated because it requires well-ordered sub-and supersolutions, which is not usually easy to get. Indeed, in many application cases, sub-and supersolutions are obtained from eigenfunction associated with the first eigenvalue of some operators (say the p-Laplacian.) But, when dealing with variable exponents, it is well known that the p(x)-Laplacian does not have in general a first eigenvalue (see [12]) and therefore, we have to find sub-and supersolution by means of other ideas (see our application example in Sect. 5). Now, we explain how this paper is organized. In Sect. 2 we introduce some notations and properties of Lebesgue-Sobolev spaces with variable exponents. Then, we give in Sect. 3 the main result, Theorem 3.2. Section 4 is devoted to prove the main result. Finally, in Sect. 5 we give an application of our main result.

Preliminaries
In this section, we briefly recall some definitions and basic properties of the generalized Lebesgue-Sobolev spaces L p(x) ( ), W 1, p(x) ( ) and W 1, p(x) 0 ( ), when is a bounded open set of R N (N ≥ 1) with a smooth boundary. For the details see [8,10,12].
We introduce the variable exponents Lebesgue space endowed with the Luxemburg norm The following inequality will be used later [8,10,12] • Moreover, for any u ∈ L p(x) ( ) and v ∈ L p (x) ( ), we have the Hölder inequality • If p + < +∞, then L p(x) ( ) is separable.
• The inclusion between Lebesgue spaces also generalizes naturally; if 0 < | | < ∞ and p 1 , p 2 are variable exponents so that p 1 (x) ≤ p 2 (x) almost everywhere in , then we have the following continuous Now, we define also the variable Sobolev space by endowed with the following norm

Hypotheses and main result
We suppose that is a bounded open set of R N (N ≥ 1) with a smooth boundary ∂ , Q = × (0, T ) where T > 0 is fixed and = ∂ × (0, T ).
endowed with the norm or, the equivalent norm The equivalence of the two norms comes from Poincaré's inequality and the continuous embedding We set We state some further properties of V 0 in the following lemma.
Lemma 3.1 [5] We denote by V 0 the dual space of V 0 . Then • We have the following continuous dense embeddings: , it is also dense in V 0 and for the corresponding dual spaces we have • One can represents the elements of V 0 as follows: let G ∈ V 0 , then there exists F = ( f 1 , f 2 , · · · , f N ) ∈ (L p (x) (Q)) N such that G = −div(F) and for any u ∈ V 0 . Now, let us give the hypotheses which concern A and f .
(A1) A is a Carathéodory function defined on Q × R × R N , with values in R N such that there exist λ > 0, and l ∈ L p (x) (Q), l ≥ 0, so that for all s ∈ R and for all ξ ∈ R N : (say growth condition of A) (A2) For all s ∈ R and for all ξ, ξ ∈ R N , with ξ = ξ : (say monotonicity condition of A ) (A3) There exists α > 0, so that for all s ∈ R and for all ξ ∈ R N : (say coercivity condition of A) (A4) f is a Carathéodory function on Q × R × R N , and there exist a function b : R + −→ R + increasing, and h ∈ L 1 (Q), h ≥ 0, such that: (say natural growth condition on f respect to |ξ | of order p(x)) , then under the assumptions (A1), (A2), and (A3) we have Au ∈ V 0 . Moreover, under the assumption (A4) we have f (x, t, u, ∇u) ∈ L 1 (Q).

Definition 3.3 A periodic solution for problem (P)
is a measurable function u : Q → R satisfying the following conditions Thanks to the previous remark and (3.2), we have ∂ t u ∈ V 0 + L 1 (Q). Moreover, the periodicity condition (3.3) makes sense according to the following lemma.
Then, we have the following embedding Now, we can ensure that all terms of (P) have a meaning.
A supersolution of problem (P) is obtained by reversing the inequalities.
We can now state the main result of this paper.

Theorem 3.6 Suppose that A verifies the hypotheses A1), A2), A3)
, and that f satisfies A4). Moreover, assume the existence of a subsolution ϕ, and a supersolution ψ, such that ϕ ≤ ψ a.e. in Q. Then, there exists at least one periodic solution u of problem (P), such that ϕ ≤ u ≤ ψ a.e. in Q.
Before we start the proof, we give some technical lemmas which will be used later.
Lemma 3.8 [1] Assume that (A1), (A2), and (A3) are satisfied and let (u n ) be a sequence in V 0 which converges weakly to u in V 0 , and Then, 4 Proof of Theorem 3.6

Truncation of problem (P)
Let ϕ be a subsolution and ψ a supersolution of problem (P), such that ϕ ≤ ψ a.e. in Q. Let us define for u ∈ V the truncation function T (u) by We shall denote by the Nemyskii operators associated, respectively, with the functions A and f .
Note that the F is not a Carathéodory function since it is not continuous with respect to ∇u. This constraint will be overcome thanks to the following lemma.
where h is a nonnegative function in L 1 (Q).
Proof The proof of this lemma is similar to the case when p(x) is a constant (see [14].) Denote by A u = −div(A (u, ∇u)); then A is a Leray-Lions's type operator from V 0 into its dual V 0 , that means A satisfies the assumptions A1), A2) and A3).

Penalization and regularization of problem (P)
We set Then, K is a closed convex set of V 0 . For u ∈ V , we define Let η > 0, the penalization operator related to K is defined by 1 η β(u). Moreover, it is clear that Let > 0, for u ∈ V , and for almost everywhere (x, t) in Q we set , and from (4.1) we can easily verify that where C is a constant which is independent of . We will now consider the following penalized-regularized problem (T ) in .
By application of Theorem 1.2, p.319 in [16] we can ensure the existence of a solution of problem (P η, ). Indeed, we have:

coercive and pseudo-monotone. Moreover, there exists at least one solution (u η, ) of problem (P η, ).
Proof Boundedness of N . From the assumption A1) and the definition of N , we have We treat each integral in the right-side member of (4.3).
By using Hölder's inequality, we get Since |F | ≤ 1 and V 0 → L 1 (Q), then we get Moreover, we have Then, by using Hölder's inequality in (4.4), we obtain Similarly, we obtain Whence, where γ is a positive constant.
Coercivity of N . From the definition of N , we have

Furthermore, we have
Hence, Pseudo-monotonicity of N . Let (u n ) ∈ D and u ∈ D, such that u n converges weakly to u in V 0 (then u n converges weakly to u in L p − (0, T ; W 1, p(x) 0 ( )), see Lemma 3.1), and ∂ t u n converges weakly to ∂ t u in V 0 (then ∂ t u n converges weakly to ∂ t u in L ( p + ) (0, T ; W −1, p (x) ( )) see Lemma 3.1 again).
Moreover, we suppose that lim n→∞ sup N u n , u n − u ≤ 0. (4.5) We shall prove that We We have the following embeddings where B 0 c → B means that B 0 is compactly embedded in B. By a theorem of Aubin-Lions's, pp. 57-58 in [16], we deduce that u n converges strongly to u in L p − (0, T ; L p(x) ( )), which embedded into L p − (Q).
Furthermore, we have By using Hölder's inequality and the embedding of V 0 into L p − (Q) in (4.8), we get Since (u n ) is bounded in V 0 and u n converges strongly to u in L p − (Q), we obtain 1 η Q |β(u n )||u n − u| → 0 when n → +∞. Now, thanks to Lemma 3.8, we get u n → u strongly in V 0 that means ∇u n → ∇u strongly in (L p(x) (Q)) N .

A-priori-estimates
In this section, we are going to obtain some estimations on the sequence solutions (u η, ) of problem (P η, ) independently of η and .

Lemma 4.4
The sequence (∂ t u η ) η is bounded in V 0 .
Proof Let v ∈ V 0 , from the first equation of problem (P η, ), we get Thus, (4.14) We treat each integral in the right-hand side of (4.14). We claim first that A (u η , ∇u η ) is bounded in for all a, b ≥ 0 and p > 1, then according to the assumption A1), we get By the inequality (2.1) for the third integral in the right-hand side of (4.15), and the fact that (u η ) is bounded in V 0 (by Lemma 4.3), we can deduce the boundedness of A (u η , ∇u η ) in (L p (x) (Q)) N . Lemma 4.3) and v ∈ V 0 → L p − (Q) → L 1 (Q), then we use Hölder's inequality in (4.14), to obtain the desired result.
As in (4.7), by Aubin-Lions's theorem, we can extract a subsequence, still denoted by (u η ) which is relatively compact in L p − (0, T ; L p(x) ( )) → L p − (Q). Furthermore, there exists u ∈ V 0 such that: for all > 0 fixed, we have as η → 0 u η → u strongly in L p − (Q) and a.e. in Q, Now, as F (u η , ∇u η ) and 1 η β(u η ) are bounded in L ( p − ) (Q), independently of η, then there exist β and (4.19) and In addition, as The estimations in V 0 and L ∞ (Q) obtained above do not allow us to pass directly to the limit in the problem (P η, ), because we can not pass to the limit in the term F (u η , ∇u η ) (which is bounded only in L 1 (Q), see (4.2).) To overcome this difficulty we need the strong convergence in V 0 of the sequence solutions (u η ). To this end, we shall prove the following lemma.

Lemma 4.5 (u η ) converges strongly to (u ) in V 0 , when η tends to zero.
Proof The proof is almost the same as in the case when the exponents p(x) = p is a constant(see [14]). Thus, we give here only a sketch. The general idea is to use Lemma 3.8, since A satisfies the hypothesis A1), A2), A3), and the weak convergence of u η to u in V 0 . Hence, it suffices to show that lim sup We consider μ > 0 and we subtract (P η, ) from (P μ, ), we get We multiply this equation by u η − u μ , and use Lemma 3.7, to obtain Firstly, we take the lim sup when η tends to 0 and secondly the lim sup when μ tends to 0 in (4.23). By using On the other hand, since u η converges to u a.e. in Q, by assumption A1), we get Moreover, from (4.16) and (4.17), we obtain Finally, we use (4.24) and (4.25) to obtain (4.22). Now, since the mapping u → F (u, ∇u) is continuous from V into L 1 (Q), the previous lemma permits to pass to the limit in the term F (u η , ∇u η ) which converges to F (u , ∇u ) in L 1 (Q). Moreover, we can also deduce the strong convergence of A u η to A u in V 0 .
Furthermore, since 1 η β(u η ) is bounded in L ( p − ) (Q), and u η converges strongly to u in V 0 , then β(u ) = 0 a.e. in Q, which implies that u is in K . Thus, u is in L ∞ (Q), this is a fundamental difference with u η (the role of the penalty operator 1 η β(u η ).) Finally, we pass to the limit in (P η, ), when η tends to zero, to obtain the following problem and one can easily deduce that

Estimates on (u ) in V 0
At this stage, we got a nonlinear problem (P ) which only depends on the parameter . So, to pass to the limit when tends to zero, we need some a priori estimates in V 0 .

Lemma 4.6
The sequence (u ) is bounded in V 0 .
Proof We prove this result by using the test function z s (u ) = exp(su 2 )u , where s is such that where α is defined in A3) and C in (4.2). As u is in By multiplying (P ) by z s (u ), we obtain From the periodicity condition of u , the first term in the left-hand side of (4.28) equals zero. We use (4.16), (4.19) and the sign condition of β, we get Q β z s (u ) ≥ 0. Moreover, by (4.2), the coercivity assumption A3), and the fact that u is in K , we obtain Now, by the (4.27) and the inequality (2.1), we get where C is independent of . Hence, (u ) is bounded in V 0 .

Lemma 4.7 The sequence (∂ t u ) is bounded in V
To prove this lemma, it suffices to show from the problem (P ) that β is bounded in L 1 (Q). In other words, we need the following estimate, whose proof is similar to that in [14, p. 296 where C 1 is independent of η and , and where C is defined in (4.2).
, then from the equation of problem (P ), we have In a similar way as in the proof of Lemma 4.4, and since (u ) is bounded in V 0 , we obtain We use (4.2), inequality (2.1) and the boundedness of (u ) in V 0 , to obtain Now, by using (4.29) and since v ∈ L ∞ (Q), we obtain Finally, we have Passage to the limit in .
We fix s > N 2 + 1, so that H s 0 ( ) → L ∞ ( ), and then L 1 ( ) → H −s ( ). We have also, ( )) ). Thus, from the previous lemma (∂ t u ) is bounded in L 1 (0, T ; H −s ( )). Moreover, from the compactness theorem of [19] (p. 85, Corollary 4) and (4.7), the sequence (u ) is relatively compact in L p − (Q). So, we can extract a subsequence still denoted by (u ), such that, when tends to zero we have u → u strongly in L p − (Q), and a.e. in Q, In addition, by using (4.31), it is clear that u is in K .

Lemma 4.8
The sequence (u ) converges strongly to some u in V 0 .
Proof The idea of proof is to apply the Lemma 3.8, since u converges weakly to u in V 0 and A satisfies A1), A2) and A3). We consider > 0 and we subtract (P ) from (P ), we obtain Now, we multiply (4.35) by the same type of test function z s (u − u ) used in the proof of Lemma 4.6, we get Thanks to the periodicity condition of u , the first term of (4.36) equals zero. By (4.16), (4.19) and the sign condition of β, the last term of (4.36) is nonnegative. By (4.2), the Eq. (4.36), then implies that Using the coercivity condition A3), we get By condition (4.27), we deduce that Following the same steps of Lemma 4.5, we obtain the desired result, namely Now, we prove that u is between ϕ and ψ almost everywhere in Q, where ϕ and ψ are, respectively, suband supersolution of problem (P) with ϕ ≤ ψ a.e. in Q. Proposition 4. 9 We have ϕ ≤ u ≤ ψ a.e. in Q.
Proof We shall prove that ϕ ≤ u a.e. in Q. One can verify easily that: v = u + (ϕ − u ) + is in K . Then, we can take it as a function test in (4.26). Hence, we obtain Since ϕ is a subsolution, and By subtracting (4.40) from (4.41), and by Lemma 3.7, we get Thanks to Lemma 4.8, we pass to the limit when tends to zero in (4.42) and get Furthermore, from the definition of A and F , we have Therefore, we obtain According to A2), this implies that ∇(ϕ − u) = 0 a.e. in {(x, t) ∈ Q, ϕ ≥ u}. Then, ϕ − u = 0 a.e. in {(x, t) ∈ Q, ϕ ≥ u} which means that ϕ ≤ u a.e. in Q. By a similar proof, we can obtain u ≤ ψ a.e. in Q.
To complete the proof of Theorem 3.6 we need the following lemma.

Conclusion
Now, we can pass to the limit in each term of problem (P ). In other words, we have Therefore, u satisfies ∂ t u + A (u, ∇u) − F (u, ∇u) = 0.
Finally, u is a periodic solution of problem (P).

Applications
In this section, we construct a subsolution and a supersolution for the following nonlinear parabolic problem associated with p(x)-Laplacian (concerning their physical interpretation see our introduction or [2] for more details): where ≡ B(0, R) = {x ∈ R N | |x| < R} is the unit ball, with R > 0 large enough. Moreover, assume that p(x) ∈ C 1 (R N ) is radial, that means p(x) = p(|x|) = p(r ), with |x| = r < R, and satisfies the assumptions of Sect. 2.