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

Transcription

1 Arab. J. Math. (27) 6: DOI.7/s Arabian Journal of Mathematics A. El Hachimi S. Maatouk Existence of periodic solutions for some quasilinear parabolic problems with variable exponents Received: 2 July 26 / Accepted: August 27 / Published online: 9 August 27 The Author(s) 27. This article is an open access publication Abstract 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. Mathematics Subject Classification 35B 35K 35K59 Introduction Let be a bounded open set of R N (N ) with a smooth boundary,andfixedt >. Our aim here is to prove the existence of periodic solutions for the following nonlinear parabolic problem t u + Au = f (x, t, u, u) in (, T ), (P) u = on (, T ), u() = u(t ) in, where Au = div(a(,, u, u)) is a Leray Lions s type operator with variable exponents acting from some functional space V into its topological dual V 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,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 [,2]) in order that the Poincaré s inequality and the density of smooth functions in W,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 A. El Hachimi (B) S. Maatouk Department of Mathematics, Faculty of Sciences, Mohammed V University, Rabat, Morocco aelhachi@yahoo.fr

2 264 Arab. J. Math. (27) 6: constant, it is well known that L p (, T ; W,p ( )) can be taken as a space of solutions. However, when p(x) is nonconstant, then nor L p(x) (, T ; W,p(x) ( )) neither L p (, T ; W,p(x) ( )), wherep = 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 as it was done by Bendahmane in [5]. Nonlinear problems defined by ( P) arise in many applications; for instance, in electrorheological fluids (see [8]), 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 [7,2]. 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 > and k(x, t) L +δ ( (, T )), c being a positive constant. In [4], N. Grenon extends the result of [9] tothe 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 [6,7]. 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 [4] were extended by El Hachimi and Lamrani in [], where the authors obtained the existence of periodic solutions, under the same hypotheses as in [4]. For variable exponents,thiskindof problemshas been studiedby many authors [2,5,3,2], 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 [] 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 [2]) 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. 2 Preliminaries In this section, we briefly recall some definitions and basic properties of the generalized Lebesgue Sobolev spaces L p(x) ( ), W,p(x) ( ) and W,p(x) ( ), when is a bounded open set of R N (N ) with a smooth boundary. For the details see [8,,2]. Let p : [, + ) be a continuous, real-valued function. Denote by p = min x p(x) and p + = max x p(x). We introduce the variable exponents Lebesgue space { } L p(x) ( ) = u : R; u is measurable with u(x) p(x) dx <, endowed with the Luxemburg norm } u L p(x) ( ) {λ> = inf : u(x) p(x) λ dx.

3 Arab. J. Math. (27) 6: The following inequality will be used later { } { } min u p L p(x) ( ), u p + u(x) p(x) dx max L p(x) ( ) u p L p(x) ( ), u p + L p(x) ( ). (2.) Lemma 2. [8,,2] (L p(x) ( ), L p(x) ( )) is a Banach space. If p >, thenl p(x) ( ) is reflexive and its conjugate space can be identified with L p (x) ( ), where p(x) + p (x) =. Moreover, for any u L p(x) ( ) and v L p (x) ( ), we have the Hölder inequality uv dx ( + ) p (p u ) L p(x) ( ) v L p (x) ( ) 2 u L p(x) ( ) v L p (x) ( ). If p + < +,thenl p(x) ( ) is separable. The inclusion between Lebesgue spaces also generalizes naturally; if < < and p,p 2 are variable exponents so that p (x) p 2 (x) almost everywhere in, then we have the following continuous embedding L p 2(x) ( ) L p (x) ( ). Now, we define also the variable Sobolev space by endowed with the following norm W,p(x) ( ) ={u L p(x) ( ); u L p(x) ( )}, u W,p(x) ( ) = u L p(x) ( ) + u L p(x) ( ). Definition 2.2 The variable exponents p : [, + ) is said to satisfy the log-hölder continuous condition if x, y, x y <, p(x) p(y) <ω( x y ), where ω : (, ) R is a nondecreasing function with lim sup α ω(α)ln ( α ) <. Lemma 2.3 [8,,2] If < p p + <, then the space (W,p(x) ( ), W,p(x) ( )) is a separable and reflexive Banach space. If p(x) satisfies the log-hölder continuous condition, then C ( ) is dense in W,p(x) ( ). Moreover, we can define the Sobolev space with zero boundary values, W,p(x) ( ) as the completion of C ( ) with respect to the norm W,p(x) ( ). For all u W,p(x) ( ), the Poincaré inequality u L p(x) ( ) C u L p(x) ( ), holds. Moreover, u,p(x) W ( ) = u,p(x) L p(x) ( ) is a norm in W ( ). Throughout this paper, we shall assume that the variable exponents p(x) satisfy the log-hölder condition and that < p p + <. 3 Hypotheses and main result We suppose that is a bounded open set of R N (N ) with a smooth boundary, = (, T ) where T > isfixedand = (, T ).

4 266 Arab. J. Math. (27) 6: Let endowed with the norm V ={f L p (, T ; W,p(x) ( )); f L p(x) ()}, f V = f L p(x) (), or, the equivalent norm f V = f L p (,T ;W,p(x) ( )) + f L p(x) (). The equivalence of the two norms comes from Poincaré s inequality and the continuous embedding L p(x) () L p (, T ; L p(x) ( )). We set V ={f L p (, T ; W,p(x) ( )); f L p(x) ()}. We state some further properties of V in the following lemma. Lemma 3. [5] We denote by V the dual space of V.Then We have the following continuous dense embeddings: L p + (, T ; W,p(x) ( )) d d V L p (, T ; W,p(x) ( )). In particular, since D() is dense in L p +(, T ; W,p(x) ( )), it is also dense in V and for the corresponding dual spaces we have L (p ) (, T ; (W,p(x) ( )) ) V L(p +) (, T ; (W,p(x) ( )) ). One can represents the elements of V as follows: let G V, then there exists F = ( f, f 2,, f N ) (L p (x) ()) N such that G = div(f) and T G, u V,V = F udxdt, for any u V. Now, let us give the hypotheses which concern A and f. (A) A is a Carathéodory function defined on R R N, with values in R N such that there exist λ>, and l L p (x) (), l, so that for all s R and for all ξ R N : (say growth condition of A) A(x, t, s,ξ) λ(l(x, t) + s p(x) + ξ p(x) ), a.e. in. (A2) For all s R and for all ξ,ξ R N, with ξ = ξ : (say monotonicity condition of A ) (A(x, t, s,ξ) A(x, t, s,ξ )) (ξ ξ )>, a.e. in. (A3) There exists α>, so that for all s R and for all ξ R N : (say coercivity condition of A) A(x, t, s,ξ) ξ α ξ p(x), a.e. in. (A4) f is a Carathéodory function on R R N, and there exist a function b : R + R + increasing, and h L (), h, such that: (say natural growth condition on f respect to ξ of order p(x)) f (x, t, s,ξ) b( s )(h(x, t) + ξ p(x) ), for (x, t, s,ξ) R R N. Remark 3.2 If u V L (), then under the assumptions (A), (A2), and(a3) we have Au V. Moreover, under the assumption (A4) we have f (x, t, u, u) L ().

5 Arab. J. Math. (27) 6: Definition 3.3 A periodic solution for problem (P) is a measurable function u : R satisfying the following conditions u V L (), t u V + L (), (3.) t u,φ V +L (),V L ()+ A(x, t, u, u) φ = f (x, t, u, u)φ for all φ V L (), (3.2) u(x, ) = u(x, T ) for all x. (3.3) Thanks to the previous remark and (3.2), we have t u V + L (). Moreover, the periodicity condition (3.3) makes sense according to the following lemma. Lemma 3.4 [5] We set W := {u V ; t u V + L ()}. Then, we have the following embedding W L () C([, T ]; L 2 ( )). Now, we can ensure that all terms of (P) have a meaning. Definition 3.5 A subsolution (in the distributional sense) of problem (P) is a function ϕ V L () such that t ϕ V + L () and t ϕ + Aϕ f (x, t,ϕ, ϕ) in, ϕ on, ϕ() ϕ(t ) in, 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 A), 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. Then, there exists at least one periodic solution u of problem (P), such that ϕ u ψ a.e. in. Before we start the proof, we give some technical lemmas which will be used later. Lemma 3.7 [5] Let π : R R be a C piecewise function with π() = and π = outside a compact set. Let (s) = s π(σ)dσ.ifu V L () with t u V + L (), then T t u,π(u) = t u,π(u) V +L (),V L () = (u(t ))dx (u())dx. Lemma 3.8 [] Assume that (A), (A2), and (A3) are satisfied and let (u n ) be a sequence in V which converges weakly to u in V, and lim sup (A(x, t, u n, u n ) A(x, t, u n, u)) ( u n u). n Then, u n u strongly in V. 4 Proof of Theorem Truncation of problem (P) Let ϕ be a subsolution and ψ a supersolution of problem (P), such that ϕ ψ a.e. in. Let us define for u V the truncation function T (u) by

6 268 Arab. J. Math. (27) 6: T (u) = u (u ψ) + + (ϕ u) +. We shall denote by A(u, u)(x, t) = A(x, t, u(x, t), u(x, t)) and F(u, u)(x, t) = f (x, t, u(x, t), u(x, t)), the Nemyskii operators associated, respectively, with the functions A and f. For almost everywhere (x, t) in, wedefine A (u, u)(x, t) = A(Tu, u)(x, t), and F (u, u)(x, t) = F(Tu, Tu)(x, t). 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. Lemma 4. The operator F : u F (u, u) is defined and continuous from V into L (). Moreover, there exists a constant C > such that F (u, u)(x, t) C(h (x, t) + u p(x) ), a.e. in, (4.) where h is a nonnegative function in L (). Proof The proof of this lemma is similar to the case when p(x) is a constant (see [4].) Denote by A u = div(a (u, u)); thena is a Leray Lions s type operator from V into its dual V, that means A satisfies the assumptions A), A2) and A3). 4.2 Penalization and regularization of problem (P) Let k > be a constant such that k ϕ ψ + k a.e. in. We set K ={v V, k v k a.e. in }. Then, K is a closed convex set of V.Foru V,wedefine β(u) =[(u k) + ] (p ) [(u + k) ] (p ). Let >, the penalization operator related to K is defined by β(u). Moreover, it is clear that β(u)u a.e. in, and K {v V,β(v)= a.e. in }. Let ɛ>, for u V, and for almost everywhere (x, t) in we set Fɛ (u, u)(x, t) = F (u, u)(x, t) + ɛ F (u, u)(x, t). It is clear that Fɛ (u, u) L () for all u V. Moreover, the mapping u Fɛ (u, u) is continuous from V into L (), and from (4.) we can easily verify that Fɛ (u, u)(x, t) C(h (x, t) + u p(x) ), a.e. in. (4.2) where C is a constant which is independent of ɛ. We will now consider the following penalized regularized problem u,ɛ V, t u,ɛ V (P,ɛ ) t u,ɛ + A u,ɛ Fɛ (u,ɛ, u,ɛ ) + β(u,ɛ) = in, u,ɛ = in, u,ɛ () = u,ɛ (T ) in. By application of Theorem.2, p.39 in [6] we can ensure the existence of a solution of problem (P,ɛ ). Indeed, we have:

7 Arab. J. Math. (27) 6: Proposition 4.2 Let D ={u V, suchthat t u V andu() = u(t )}. Then, the operator N u = A u Fɛ (u, u)+ β(u) defined from D into V is bounded, coercive and pseudo-monotone. Moreover, there exists at least one solution (u,ɛ ) of problem (P,ɛ ). Proof Boundedness of N. From the assumption A) and the definition of N,wehave ( ) N u,v λ l(x, t) v + Tu p(x) v + u p(x) v + Fɛ (u, u) v + β(u) v. (4.3) We treat each integral in the right-side member of (4.3). By Remark 3.2 (recall that ϕ,ψ are in the L ()), we have l(x, t) v + Tu p(x) v C v V. By using Hölder s inequality, we get u p(x) v c 2 u p(x) L p (x) () v V. Then, if u p(x) L p (x) (), it s over. Otherwise, from inequality (2.), we have ( u p(x) (p ) L p (x) () = u p(x) p + u L p (x) () p(x) ) p (x), and ( u p(x) ) p (x) { } { } max u p L p(x) (), u p + = max u p L p(x) () V, u p + V. Hence, { } p u p(x) p + v c max u V, u (p +) v V. Since Fɛ ɛ and V L (), thenweget Fɛ (u, u) v (c /ɛ) v V. Moreover, we have β(u) v ( V ) (u k) + (p ) v + (u + k) (p ) v. (4.4) Now, since u V L p (),weget((u k) + ) (p ) L (p ) (). Then, by using Hölder s inequality in (4.4), we obtain (u k) + (p ) v c 3 (u k) + (p ) V v V c 3 u (p ) V v V. Similarly, we obtain Whence, where γ is a positive constant. (u + k) (p ) v c 4 u (p ) V v V. ( { }) p N u V γ + u (p ) p + V + max u V, u (p +) V.

8 27 Arab. J. Math. (27) 6: Coercivity of N. From the definition of N,wehave N u, u = A u, u F ɛ (u, u), u + β(u), u. Furthermore, we have A u, u = Fɛ (u, u), u = and Hence, A (u, u) u α F (u, u) + ɛ F (u, u) u N u, u u V { } u p(x) α min u p V, u p + V, β(u), u = β(u)u. { α min u (p ) V F (u, u) + ɛ F (u, u) u ɛ u L () c ɛ u V,, u (p +) V } c ɛ. Whence, N u, u +, when u V +. u V Pseudo-monotonicity of N. Let (u n ) D and u D, such that u n converges weakly to u in V (then u n converges weakly to u in L p (, T ; W,p(x) ( )), see Lemma 3.), and t u n converges weakly to t u in V (then tu n converges weakly to t u in L (p +) (, T ; W,p (x) ( )) see Lemma 3. again). Moreover, we suppose that We shall prove that lim n sup N u n, u n u. (4.5) lim n inf N u n, u n v N u, u v for all v V. (4.6) Choose s > N 2 + such that W,p (x) ( ) H s ( ); then t u n converges weakly to t u in L (p +) (, T ; H s ( )). We set B = W,p(x) ( ), B = L p(x) ( ) and B = H s ( ). We have the following embeddings B c B B, (4.7) c where B B means that B is compactly embedded in B. By a theorem of Aubin Lions s, pp in [6], we deduce that u n converges strongly to u in L p (, T ; L p(x) ( )), which embedded into L p (). Furthermore, we have β(u n ) u n u [ ] (u n k) + (p ) u n u + (u n + k) (p ) u n u. (4.8) By using Hölder s inequality and the embedding of V into L p () in (4.8), we get β(u n ) u n u c ( ) u n (p ) V u n u L p ().

9 Arab. J. Math. (27) 6: Since (u n ) is bounded in V and u n converges strongly to u in L p (), weobtain β(u n ) u n u when n +. (4.9) On the other hand, since Fɛ ɛ and L p () is embedded in L (), weget Fɛ (u n, u n ) u n u c ɛ u n u L p () when n. (4.) We develop each term of N u n, u n u and use (4.5), (4.9)and(4.), to obtain lim sup N u n, u n u = lim sup A(Tu n, u n ) (u n u). (4.) n n Applying Vitali s theorem and weak convergence of A(Tu n, u) in (L p (x) ()) N,weobtain A(Tu n, u) (u n u) =. (4.2) lim n By using (4.)and(4.2), then we obtain [A(Tu n, u n ) A(Tu n, u)] ( u n u). lim n Now, thanks to Lemma 3.8,weget u n u strongly in V that means u n u strongly in (L p(x) ()) N. Hence, the inequality (4.6) desired. Finally, by Theorem.2, p. 39 in [6], we deduce the existence of at least one solution (u,ɛ ) of problem ). (P,ɛ 4.3 A-priori-estimates In this section, we are going to obtain some estimations on the sequence solutions (u,ɛ ) of problem (P,ɛ ) independently of and ɛ Estimates on (u,ɛ ) in V and L () Let us fix ɛ, and denote by (u ) : (u,ɛ ). Lemma 4.3 The sequences ( β(u ) ) and (u ) are bounded respectively in L (p ) () and V. Proof From the definition of β(u ), we deduce that β(u ) (u k) + L (p ) () Then, we only need to show that: ( ) (u k) + (p ) and (p ) (p ) L p () ( ) (u + k) (p ) (u + k) + (p ) (p ) L p () are bounded in L p (). Since (u k) + V, then by multiplying (P,ɛ ) by (u k) +,weget t u,(u k) + + A (u, u ) (u k) + Fɛ (u, u )(u k) + + ((u k) + ) p =..

10 272 Arab. J. Math. (27) 6: Since u () = u (T ), then from Lemma 3.7, we deduce that t u,(u k) + =. Hence ((u k) + ) p = Fɛ (u, u )(u k) + A (u, u ) (u k) +. (4.3) Under the assumption A3), the second integral in the right-hand side of equality (4.3) is nonnegative. On the other hand, we have Fɛ /ɛ. Let us divide both sides of equality (4.3) by/(p ). By Hölder s inequality, we obtain ((u k) + ) p [ ((u k) + ) p ] /p C, where C is indepedent on, whence ( (u k) + (p ) ) (p ) is bounded in L p (). (p ) Using ( (u + k) ) as a test function, we prove in the same way that ( (u +k) (p ) ) is bounded in L p (). Now we prove that (u ) is bounded in V. Multiplying (P,ɛ ) by u and using the assumption A3), Lemma 3.7, and the fact that β(u )u, we obtain α u p(x) A (u, u ) u Fɛ (u, u )u c ɛ u V. By using the inequality (2.), hence, we get { min u (p ) V, u (p +) V } c(ɛ, α). Since (p ) and(p + ) are strictly greater than, we deduce that (u ) is bounded in V. Lemma 4.4 The sequence ( t u ) is bounded in V. Proof Let v V, from the first equation of problem (P,ɛ ),weget t u,v = A u,v + Fɛ (u, u ), v β(u ), v. Thus, t u,v A (u, u ) v + Fɛ (u, u ) v + β(u ) v. (4.4) We treat each integral in the right-hand side of (4.4). We claim first that A (u, u ) is bounded in (L p (x) ()) N.Indeed,if A (u, u ) L p (x) (), the claim is obvious. Otherwise, we have A (u, u ) (p ) A L p (x) () (u, u ) p (x), since p (x) (p ) +,and(a + b) p 2 p (a p + b p ) for all a, b andp >, then according to the assumption A),weget [ A (u, u ) p (x) c(λ, (p ) + ) ] l(x, t) p (x) + Tu p(x) + u p(x). (4.5) By the inequality (2.) for the third integral in the right-hand side of (4.5), and the fact that (u ) is bounded in V (by Lemma 4.3), we can deduce the boundedness of A (u, u ) in (L p (x) ()) N. Since Fɛ ɛ, β(u ) is bounded in L (p ) () (by Lemma 4.3) andv V L p () L (), then we use Hölder s inequality in (4.4), to obtain the desired result.

11 Arab. J. Math. (27) 6: As in (4.7), by Aubin Lions s theorem, we can extract a subsequence, still denoted by (u ) which is relatively compact in L p (, T ; L p(x) ( )) L p (). Furthermore, there exists u ɛ V such that: for all ɛ> fixed, we have as u u ɛ strongly in L p () and a.e. in, (4.6) u u ɛ weakly in V, (4.7) t u t u ɛ weakly in V. (4.8) Now, as Fɛ (u, u ) and β(u ) are bounded in L (p ) (), independently of, then there exist β ɛ and F ɛ in L (p ) (), such that β(u ) β ɛ in L (p ) (), (4.9) and F ɛ (u, u ) F ɛ in L (p ) (). (4.2) In addition, as A (u, u ) is bounded in (L p (x) ()) N, then there exists χ ɛ in (L p (x) ()) N such that A (u, u ) χ ɛ in (L p (x) ()) N (L (p ) ()) N. (4.2) The estimations in V and L () obtained above do not allow us to pass directly to the limit in the problem (P,ɛ ), because we can not pass to the limit in the term F ɛ (u, u ) (which is bounded only in L (), see (4.2).) To overcome this difficulty we need the strong convergence in V of the sequence solutions (u ).To this end, we shall prove the following lemma. Lemma 4.5 (u ) converges strongly to (u ɛ ) in V,when tends to zero. Proof The proof is almost the same as in the case when the exponents p(x) = p is a constant(see [4]). Thus, we give here only a sketch. The general idea is to use Lemma 3.8, sincea satisfies the hypothesis A), A2), A3), and the weak convergence of u to u ɛ in V. Hence, it suffices to show that ( lim sup A (u, u ) A (u, u ɛ ) ) ( u ) u ɛ =. (4.22) We consider μ>and we subtract (P,ɛ ) from (P μ,ɛ ),weget t u t u μ + A u A u μ F ɛ (u, u ) + F ɛ (u μ, u μ ) + β(u ) μ β(u μ) =. We multiply this equation by u u μ, and use Lemma 3.7, to obtain ( A (u, u ) A (u μ, u μ ) ) ( ( u u μ ) F ɛ (u, u ) Fɛ (u μ, u μ ) ) (u u μ ) ( + β(u ) ) μ β(u ) (u u μ ) =. (4.23) Firstly,wetakethelimsupwhen tends to and secondly the lim sup when μ tends to in (4.23). By using (4.6), (4.7), (4.8), (4.9)and(4.2), we obtain lim sup A (u, u ) u χ ɛ u ɛ So, for μ =,weget χ ɛ u ɛ + lim sup A (u μ, u μ ) u μ =. μ lim sup A (u, u ) u = χ ɛ u ɛ. (4.24)

12 274 Arab. J. Math. (27) 6: On the other hand, since u converges to u ɛ a.e. in, by assumption A), weget A (u, u ɛ ) A (u ɛ, u ɛ ), strongly in (L p (x) ()) N. Moreover, from (4.6)and(4.7), we obtain A (u, u ɛ ) (u u ɛ ), when. (4.25) Finally, we use (4.24)and(4.25) to obtain (4.22). Now, since the mapping u Fɛ (u, u) is continuous from V into L (), the previous lemma permits to pass to the limit in the term Fɛ (u, u ) which converges to Fɛ (u ɛ, u ɛ ) in L (). Moreover, we can also deduce the strong convergence of A u to A u ɛ in V. Furthermore, since β(u ) is bounded in L (p ) (),andu converges strongly to u ɛ in V,thenβ(u ɛ ) = a.e. in, which implies that u ɛ is in K. Thus, u ɛ is in L (), this is a fundamental difference with u (the role of the penalty operator β(u ).) Finally, we pass to the limit in (P,ɛ ),when tends to zero, to obtain the following problem u ɛ V L (), t u ɛ V (Pɛ + L (), ) t u ɛ + A u ɛ Fɛ (u ɛ, u ɛ ) + β ɛ = in, u ɛ () = u ɛ (T ) in, and one can easily deduce that v K, t u ɛ,v u ɛ Estimates on (u ɛ ) ɛ in V A (u ɛ, u ɛ ) (v u ɛ ) Fɛ (u ɛ, u ɛ )(v u ɛ ). (4.26) At this stage, we got a nonlinear problem (Pɛ ) which only depends on the parameter ɛ. So, to pass to the limit when ɛ tends to zero, we need some a priori estimates in V. Lemma 4.6 The sequence (u ɛ ) is bounded in V. Proof We prove this result by using the test function z s (u ɛ ) = exp(su 2 ɛ )u ɛ,wheres is such that αz s (u ɛ) C z s (u ɛ ) α 2, (4.27) where α is defined in A3) and C in (4.2). As u ɛ is in V L (), thenz s (u ɛ ) is in V L (). By multiplying (Pɛ ) by z s(u ɛ ),weobtain t u ɛ, z s (u ɛ ) + A (u ɛ, u ɛ ) z s (u ɛ ) + β ɛ z s (u ɛ ) = Fɛ (u ɛ, u ɛ )z s (u ɛ ). (4.28) From the periodicity condition of u ɛ, the first term in the left-hand side of (4.28) equals zero. We use (4.6), (4.9) and the sign condition of β, weget β ɛz s (u ɛ ). Moreover, by (4.2), the coercivity assumption A3), and the fact that u ɛ is in K, we obtain ( ) α z s (u ɛ) u ɛ p(x) C + z s (u ɛ ) u ɛ p(x). Now, by the (4.27) and the inequality (2.), we get { } min u ɛ p V, u ɛ p + V 2C α, where C is independent of ɛ. Hence, (u ɛ ) is bounded in V.

13 Arab. J. Math. (27) 6: Lemma 4.7 The sequence ( t u ɛ ) is bounded in V + L (). To prove this lemma, it suffices to show from the problem (Pɛ ) that β ɛ is bounded in L (). Inother words, we need the following estimate, whose proof is similar to that in [4, p. 296] β(u,ɛ) L () C + C(h (x, t) + u,ɛ p(x) ), (4.29) where C is independent of and ɛ, andwherec is defined in (4.2). Proof Let v V L (), then from the equation of problem (Pɛ ),wehave t u ɛ,v A (u ɛ, u ɛ ) v + Fɛ (u ɛ, u ɛ ) v + β ɛ v. (4.3) In a similar way as in the proof of Lemma 4.4, and since (u ɛ ) is bounded in V, we obtain A (u ɛ, u ɛ ) v C v V. We use (4.2), inequality (2.) and the boundedness of (u ɛ ) in V, to obtain Fɛ (u ɛ, u ɛ ) v C v V. Now, by using (4.29) and since v L (), we obtain β ɛ v C v L (). Finally, we have Passage to the limit in ɛ. t u ɛ V +L () C, where C is independent of ɛ. We fix s > N 2 +, so that H s( ) L ( ), and then L ( ) H s ( ). Wehavealso, H s( ) W,p(x) ( ), and consequently, W,p (x) ( ) H s ( ). From Lemma 3. we have V L (p +) (, T ; (W,p(x) ( )) ). Thus, from the previous lemma ( t u ɛ ) is bounded in L (, T ; H s ( )). Moreover, from the compactness theorem of [9] (p. 85, Corollary 4) and (4.7), the sequence (u ɛ ) is relatively compact in L p (). So, we can extract a subsequence still denoted by (u ɛ ), such that, when ɛ tends to zero we have u ɛ u strongly in L p (), and a.e. in, (4.3) u ɛ u weak in L (), (4.32) t u ɛ t u weakly in V + L (). (4.33) Now, as A (u ɛ, u ɛ ) is bounded in (L p (x) ()) N, there exists χ in (L p (x) ()) N such that A (u ɛ, u ɛ ) χ in (L p (x) ()) N (L (p ) ()) N. (4.34) In addition, by using (4.3), it is clear that u is in K. Lemma 4.8 The sequence (u ɛ ) converges strongly to some u in V.

14 276 Arab. J. Math. (27) 6: Proof The idea of proof is to apply the Lemma 3.8, sinceu ɛ converges weakly to u in V and A satisfies A), A2) and A3). We consider ɛ > and we subtract (P ɛ ) from (P ɛ ), we obtain t (u ɛ u ɛ ) + A u ɛ A u ɛ F ɛ (u ɛ, u ɛ ) + F ɛ (u ɛ, u ɛ ) + β ɛ β ɛ =. (4.35) Now, we multiply (4.35) by the same type of test function z s (u ɛ u ɛ ) used in the proof of Lemma 4.6,weget t (u ɛ u ɛ ), z s (u ɛ u ɛ ) + + (Fɛ (u ɛ, u ɛ ) F ɛ (u ɛ, u ɛ ))z s (u ɛ u ɛ ) + (A (u ɛ, u ɛ ) A (u ɛ, u ɛ )) (u ɛ u ɛ )z s (u ɛ u ɛ ) (β ɛ β ɛ )z s (u ɛ u ɛ ) =. (4.36) Thanks to the periodicity condition of u ɛ, the first term of (4.36) equals zero. By (4.6), (4.9) and the sign condition of β, the last term of (4.36) is nonnegative. By (4.2), the Eq. (4.36), then implies that (A (u ɛ, u ɛ ) A (u ɛ, u ɛ )) (u ɛ u ɛ )z s (u ɛ u ɛ ) C (h (x, t) + u ɛ p(x) ) z s (u ɛ u ɛ ) + C (h (x, t) + u ɛ p(x) ) z s (u ɛ u ɛ ). (4.37) Using the coercivity condition A3), weget (A (u ɛ, u ɛ ) A (u ɛ, u ɛ )) (u ɛ u ɛ )z s (u ɛ u ɛ ) 2C h (x, t) z s (u ɛ u ɛ ) + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ) + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ) α α 2C h (x, t) z s (u ɛ u ɛ ) + C A (u ɛ, u ɛ ) (u ɛ u ɛ ) z s (u ɛ u ɛ ) α + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ) C A (u ɛ, u ɛ ) (u ɛ u ɛ ) z s (u ɛ u ɛ ) α α + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ). (4.38) α By condition (4.27), we deduce that (A (u ɛ, u ɛ ) A (u ɛ, u ɛ )) (u ɛ u ɛ ) 2C h z s (u ɛ u ɛ ) 2 + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ) + C A (u ɛ, u ɛ ) u ɛ z s (u ɛ u ɛ ). (4.39) α α Following the same steps of Lemma 4.5, we obtain the desired result, namely ( lim sup A (u ɛ, u ɛ ) A (u ɛ, u) ) ( u ɛ u). ɛ Now, we prove that u is between ϕ and ψ almost everywhere in, whereϕ and ψ are, respectively, suband supersolution of problem (P) with ϕ ψ a.e. in. Proposition 4.9 We have ϕ u ψ a.e. in.

15 Arab. J. Math. (27) 6: Proof We shall prove that ϕ u a.e. in. One can verify easily that: v = u ɛ + (ϕ u ɛ ) + is in K. Then, we can take it as a function test in (4.26). Hence, we obtain t u ɛ,(ϕ u ɛ ) + + A (u ɛ, u ɛ ) (ϕ u ɛ ) + Fɛ (u ɛ, u ɛ )(ϕ u ɛ ) +. (4.4) Since ϕ is a subsolution, and (ϕ u ɛ ) + in V L (), weobtain t ϕ,(ϕ u ɛ ) + + A (ϕ, ϕ) (ϕ u ɛ ) + F(ϕ, ϕ)(ϕ u ɛ ) +. (4.4) By subtracting (4.4) from (4.4), and by Lemma 3.7, weget (A(ϕ, ϕ) A (u ɛ, u ɛ )) (ϕ u ɛ ) + + (Fɛ (u ɛ, u ɛ ) F(ϕ, ϕ))(ϕ u ɛ ) +. (4.42) Thanks to Lemma 4.8, we pass to the limit when ɛ tends to zero in (4.42) and get (A(ϕ, ϕ) A (u, u)) (ϕ u) + + Furthermore, from the definition of A and F, we have (F (u, u) F(ϕ, ϕ))(ϕ u) +. (4.43) (F (u, u) F(ϕ, ϕ))(ϕ u) + = a.e.in and A (u, u) (ϕ u) + = A(ϕ, u) (ϕ u) +. Therefore, we obtain that is (A(ϕ, ϕ) A(ϕ, u)) (ϕ u) +, {ϕ u} (A(ϕ, ϕ) A(ϕ, u)) (ϕ u). According to A2), this implies that (ϕ u) = a.e.in{(x, t),ϕ u}. Then, ϕ u = a.e.in {(x, t),ϕ u} which means that ϕ u a.e. in. By a similar proof, we can obtain u ψ a.e. in. To complete the proof of Theorem 3.6 we need the following lemma. Lemma 4. β ɛ tends to zero in L (). Proof By taking (u ɛ k + ) + V L () as a test function in (Pɛ ), and using the periodicity condition of u ɛ and assumption A3), we obtain β ɛ (u ɛ k + ) + Fɛ (u ɛ, u ɛ )(u ɛ k + ) +. (4.44) On the other hand, we have β ɛ = { u ɛ <k} β ɛ + β ɛ + β ɛ. (4.45) {u ɛ =k} {u ɛ = k} The definition of β gives β(u ɛ, ) = if u ɛ, k, hence, β(u ɛ,) tends to, when tends to that is β(u ɛ,)χ { uɛ <k} a.e. in. (4.46) From (4.46), the boundedness of β(u ɛ,)χ { uɛ <k} in L (p ) () (see Lemma 4.3) and by using Lemma 4.2 of [3], we get β(u ɛ,)χ { uɛ <k} when in L (p ) () weakly. (4.47)

16 278 Arab. J. Math. (27) 6: By (4.9), we deduce that Since u ɛ < k a.e. in, then from (4.48), we obtain β ɛ (u ɛ k + ) + = So, (4.44) becomes {u ɛ =k} β ɛ = a.e. in { u ɛ < k}. (4.48) {u ɛ =k} β ɛ. β ɛ F ɛ (u ɛ, u ɛ )(u ɛ k + ) +. (4.49) Since k + ϕ u ψ k, (u ɛ k + ) + tends to almost everywhere in and in L () weak, and Fɛ (u ɛ, u ɛ ) converges strongly to F(u, u) in L (). Then, we can deduce from (4.49)that β ɛ =. lim ɛ {u ɛ =k} In the same way, we show that lim ɛ {u ɛ = k} β ɛ =. Whence, the desired result Conclusion Now, we can pass to the limit in each term of problem (Pɛ ). In other words, we have Therefore, u satisfies A (u ɛ, u ɛ ) A (u, u) in V strongly, Fɛ (u ɛ, u ɛ ) F (u, u) in L () strongly, β ɛ in L () strongly, t u ɛ t u in V + L () strongly. t u + A (u, u) F (u, u) =. From Proposition 4.9 we have ϕ u ψ. Then, we get A (u, u) = A(u, u) and F (u, u) = F(u, u). Concerning the periodicity condition, since (u ɛ ) is bounded in V L () and ( t u ɛ ) is bounded in V + L (),then( t u ɛ ) is bounded in L (, T ; H s ( )).So,(u ɛ ) is relatively compact in L p (). Hence, u ɛ () u() in L p () and u ɛ (T ) u(t ) in L p (). Asu ɛ () = u ɛ (T ), then we deduce that u() = u(t ). Finally, u is a periodic solution of problem (P). 5 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] formore details): t u p(x) u = f (x, t) in, (P) u = on, u() = u(t ) in, where B(, R) ={x R N x < R} is the unit ball, with R > large enough. Moreover, assume that p(x) C (R N ) is radial, that means p(x) = p( x ) = p(r), with x =r < R, and satisfies the assumptions of Sect. 2.

17 Arab. J. Math. (27) 6: Let M = f L () <.Weset R [ ] M ψ(r) = r N t p(t) dt, and ϕ(r) = ψ(r). It is clear that ϕ(r) ψ(r). Moreover, ψ and ϕ are supersolution and subsolution, respectively of problem (P). Indeed, we have p(r) ψ(r) = r N (r N ψ (r) p(r) 2 ψ (r)). Since ( ) M ψ (r) = N r p(r), then Now, since ψ(r) is independent of t, then we obtain ψ (r) p(r) 2 ψ (r) = M N r. t ψ(r) p(r) ψ(r) = p(r) ψ(r) = M = f L () f (x, t). Moreover, if r (ie. r = R), then ψ(r) =. Hence, ψ is a supersolution of problem (P) in the sense of Definition 3.5. We repeat the same previous calculations, to obtain t ϕ(r) p(r) ϕ(r) = p(r) ϕ(r) = M = f L () f (x, t), as far as ϕ(r) = ifr. Hence, ϕ is a subsolution of problem (P) in the sense of Definition 3.5. Hence, applying our main result, Theorem 3.6, we deduce the existence of at least one periodic solution u(x, t) of problem (P) such that ϕ u ψ a.e. in. Acknowledgements The authors would like to thank the anonymous referees for their helpfull remarks and comments. Open Access This article is distributed under the terms of the Creative Commons Attribution 4. International License ( creativecommons.org/licenses/by/4./), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References. Aharouch, L.; Azroul, E.; Rhoudaf, M.: Strongly nonlinear variational parabolic problems in weighted Sobolev spaces. Aust. J. Math. Anal. Appl 5(2), 25 (28). Art Akagi, G.; Matsuura, K.: Well-posedness and large-time behaviors of solutions for a parabolic equation involving p(x)- Laplacian. In: The eighth international conference on dynamical systems and differential equations, a supplement volume of discrete and continuous dynamical systems, pp (2) 3. Azroul, E.; Benboubker, M.B.; Redwane, H.; Yazough, C.: Renormalized solutions for a class of nonlinear parabolic equations without sign condition involving nonstandard growth. Ann. Univ. Craiova Math. Comput. Sci. Ser. 4(), 9 (24) 4. Bendahmane, M.; Wittbold, P.: Renormalized solutions for a nonlinear elliptic equations with variable exponents and L -data. Nonlinear Anal. 7, (29) 5. Bendahmane, M.; Wittbold, P.; Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and L -data. J. Differ. Equ. 249, (2) 6. Boccardo, L.; Murat, F.; Puel, J.P.: Existence results for some quasilinear parabolic equations. Non Lin. Anal. TMA 3, (988) 7. Chen, Y.; Levine, S.; Rao, M.: Variable exponents, linear growth functionals in image restoration. SIAM. J. Appl. Math. 66, (26) 8. Cruz-Uribe, D.V.; Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Birkhäuser, Basel (23) 9. Deuel, J.; Hess, P.: Nonlinear parabolic boundary value problems with upper and lower solutions. Isr. J. Math. 29, 92 4 (978)

18 28 Arab. J. Math. (27) 6: Diening, L.; Harjulehto, P.; Hästo, P.; Růži cka, M.: Lebesgue and Sobolev spaces with variable exponents, volume 27 of Lecture Notes in Mathematics. Springer, Heidelberg (27). El Hachimi, A.; Lamrani, A.A.: Existence of stable periodic solutions for quasilinear parabolic problems in the presence of well-ordered lower and uppersolutions. EJDE 9, (22) 2. Fan, X.; Zhao, D.: On the spaces L p(x) ( ) and W,p(x) ( ). J. Math. Anal. Appl. 263, (2) 3. Fu, Y.; Pan, N.: Existence of solutions for nonlinear parabolic problem with p(x)-growth. J. Math. Anal. Appl. 362, (2) 4. Grenon, N.: Existence results for some quasilinear parabolic problems. Ann. Mat. Pura Appl. 4, (993) 5. Li, Z.; Yan, B.; Gao, W.: Existence of solutions to a parabolic p(x)-laplace equation with convection term via L estimates. EJDE 25(46), 2 (25) 6. Lions, J.-L.: uelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Paris (969) 7. Mokrane, A.: Existence of bounded solutions of some nonlinear parabolic equations. Proc. R. Soc. Edinb. 7, (987) 8. Rajagopal, K.; Růži cka, M.: On the modeling of electrorheological materials. Mech. Res. Commun. 23(4), 4 47 (996). 3, Simon, J.: Compact sets in the space L p (, T ; B). Ann. Mat. Pura Appl. (IV) 46, (987) 2. Zhang, C.; Zhou, S.: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L data. J. Differ. Equ. 284, (2) 2. Zhikov, V.V.: On the density of smooth functions in Sobolev-Orlicz spaces. Zap. Nauchn. Sem. S.Peterbg. Otdel. Mat. Inst. Steklov. (POMI) 3, 67 8 (24)