Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian
This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.
The investigation of the asymptotic behavior for nonlinear parabolic equations involving the so-called p-Laplacian operator has been addressed by several authors in the last decades, in both bounded and unbounded domains, with constant or variable exponents (see [1–8]). One way to treat this question is to analyze the existence and structure (regularity and finite or infinite dimensionality of the attractor generated by the solutions of the governed equation (see ). The existence of the global attractor for the related semigroup acting on the natural weak energy space has been proved in [7, 10, 11].
In this paper, our goal is to study the time discretization for a doubly nonlinear parabolic equation associated with the p (x)-Laplacian, where in addition to usual questions of existence, uniqueness, and stability of the solutions, we will be concerned with the existence of absorbing sets and the global attractor as well. The problem under consideration is of the formwhere , with , is a nonlinearity of porous media type, is a nonlinearity of reaction type, and is an open bounded set of with smooth boundary.
Existence results and qualitative properties concerning the solutions of the continuous problem (1) and more general problems have been obtained by many authors in the last decade. We cote the papers [1–6, 12] and the references therein.
Our motivation to study problem (1) is the fact that it is considered in particular as a model of an important class of non-Newtonian fluids which are well known as electrorheological fluids (see ).
This paper is organized as follows: In Section 2, we give some preliminaries and notation. In Section 3, we discretize problem (1) by using Euler-forward scheme and obtain existence, uniqueness, and stability results.
Finally, in Section 4 we show the existence of absorbing sets in , which in turn ensures the existence of a compact global attractor.
We begin with a review of some basic results that will be needed in the subsequent sections. The known results are stated without proofs. We shall however provide references where the proofs can be found.
We first introduce the space and and state some of their properties.
Let stand for a regular open bounded set of and be a measurable bounded function as a variable exponent. Denote We define the variable exponent Lebesgue space by endowed with the Luxembourg norm The following results can be found in [14–17].
Lemma 1. Let be a measurable function with . Then, we have for all .
Proposition 2. The space is a separable, uniform convex Banach space, and its conjugate space is , where . Moreover, for any and , we have
Let denote the space of measurable functions such that and the distributional derivative are in . The norm makes a Banach space.
LetWe say that satisfies the log-Hölder condition in if where satisfies It is well known that if satisfies the log-Hölder condition (9), then the space is dense in . Moreover, we can define the Sobolev space with zero boundary values, as the completion of , with respect to the norm .
Let us recall the following versions of Poincaré’s inequality.
Lemma 3. If is continuous in , then there exists a constant such that for all and thus and are equivalent norms in .
Let us next consider the modular version of Poincaré’s inequality.
Lemma 4. Let be an element of and let . There exists a constant depending only on such that
3. The Semidiscretized Problem: Existence, Uniqueness, and Stability
Let be a continuous increasing function with . For , we set We consider the following Euler-forward scheme associated with (1):where , with being a fixed positive real, and We shall be concerned with the following two cases: or .
3.1. Case 1:
We assume the following hypotheses:
the function is an increasing and continuous from to such that for any with .
for , the map is measurable and, a.e. in , is continuous. Furthermore, we assume that there exists , such that, for a.e. , we have .
there exists , such that, for almost , is increasing.
Lemma 5. Assume and . Then, for all , we have .
Theorem 6. Assume , , and . For , there exists a unique solution of (14) in provided that .
Proof. We can write (14) as By using , , and applying Theorem 4.3 of  and Lemma 5, we deduce the existence of at least one solution for .
Let us now prove the uniqueness. For simplicity, we set Then, problem (14) readsIf and are two solutions of (14), thenMultiplying (19) by and integrating over give where denotes the pairing between and .
Then, applying yieldsNow by using (21) and the monotonicity of the p(x)-Laplacian operator, (20) reduces toThen by , we get for .
Theorem 7. Assume and . Then, there exists a constant , depending on , and , but not on , such that, for all ,
Proof. (i) From Lemma 5, we have . Then, multiplying (14) by and integrating over , we getSince and and are monotone, then we have Therefore, we obtainHence,By simple induction, we getFinally, as , we obtain (23).
(ii) In order to prove (24), we multiply (14) by (with k instead of n). By using , we get Thanks to the properties of the Legendre transformation, we get Then, we haveFinally, after summation of (33) from k=1 to n, we deduce thatWe set Then, the continuity of and the use of Lemma 5 allow us to conclude to the proof of point (24).
(iii) To prove point (25), we multiply the first equation of (17) by . By using , we getWith the aid of the elementary identity, for any reals and , we get from (36) thatNow, we take the sum of (38) from to to obtainThus, by and Lemma 5 we deduce (25).
Lemma 8. For all , there exists a positive constant depending either on or such that, for , we have
Proof. If , for any , then we have the following inequality for any : By setting and and integrating over , we get Then, by Holder’s inequality we get Let and and be such that and Then, we get Therefore, we have Hence, by (24) of Theorem 7 we get the desired result.
Lemma 9. Assume . Then, for all , we have
Proof. As , for any , then we have the following inequality for any : By setting and and integrating over , we get Hence
We can also derive a uniqueness result for problem (17) if we replace by the following hypothesis:
for all , there exists such that, if , then where with being a positive constant to be prescribed below.
Proposition 10. Assume , , and . Then, problem (14) has a unique solution for all , where is a prescribed constant.
Proof. Let and be two solutions of (14).
First case: suppose that , for all . Then, from (20) and by using Lemma 8 and Holder’s inequality, we getLet be such that andThen, by , , and , Lemma 4, and Young’s inequality, we getTherefore, for , we get .
Second case: suppose that , for all . From (20) and by using Lemma 9 and Young’s inequality, we getThen, by using and we get Thus, from Lemma 4 we get Hence, when we have .
3.2. Case 2:
Theorem 11. Assume that , , and hold true. Then, for , there exists a unique solution of (14) in provided that where is some positive constant.
Proof. The proofs of existence and uniqueness are the same as those of Theorem 6. Therefore, we omit them.
Now, we consider the following assumption:
for any , the map is measurable and, a.e. in , is continuous. Furthermore, we assume that there exist with and positive constants and such that Then, we have the following stability theorem.
Theorem 12. Assume that and are fulfilled. Then, there exists a constant such that, for all ,where and are two constants each depending either on or on .
Proof. Since the proof is nearly the same as that of Theorem 7, we just sketch it.
The argument that allowed us to get (34), with , allows also us to writeBy using Lemmas 4 and 5, and Young’s inequality, we get that for all there exists such thatSince is positive then, for a suitable choice of , we infer from (66) thatBy taking and such that and using (66) and (67), we deduce that As in (39), by using , we get Hence, by , (67), and Lemma 5, we deduce (64).
4. Absorbing Sets in : Existence of the Attractor
In this section we consider the following problems: for all integer with and fixed such that where .
We assume that , , and hold true in all the remaining section.
The result of Theorem 6 on the existence and uniqueness of the solution of (14) allows us to define a map on by setting Since is continuous, we have The existence of an absorbing set in allows us to prove the existence of a global compact attractor (see ). This will be done next in Theorem 14.
Proposition 13. If satisfies then there is an absorbing set in . Namely, for any there exists such that
Proof. We multiply (14) by . We obtainLet us denote By , is a convex function and hence satisfies the standard inequality Consequently, Now, by using , we get that is a convex function and hence we have Thus, we obtain The following inequality holds, for any and in : By setting and and integrating over , we get Now, since , then from (75), we deduce that On the other hand, by writing where , we haveDenote the left hand side of (85) by . By using and relations (23) and (24) of Theorem 7 and taking , we deduce that there exists such that Then, by applying the discrete version of the uniform Gronwall Lemma (see Lemma 7.5 of ) with , we obtainThus by Lemma 5, we deduce thatTherefore, from (88) and Theorem 7, we conclude to the desired relation
Now we are able to state our result on the existence of a compact attractor.
Theorem 14. Suppose that . Then, for , the discretized problem (71) has an associated semigroup solution that maps into . This semigroup has a compact attractor which is bounded in .
Proof. The nonlinear map defines a semigroup from into . By Proposition 13, the existence of an absorbing ball in is guaranteed.
We define the set of as Then, by the results of Temam (see ), is a compact attractor which attracts all bounded sets of , which means that, for all , we have
The authors declare that no data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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