Abstract
We consider the following nonlinear parabolic equation: , where and the exponent of nonlinearity are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.
1. Introduction
Let be a bounded domain in with a smooth boundary . We consider the following initial and boundary value problem:where and are given functions. The exponent is a given measurable function on such thatwithWe also assume that satisfies the log-Hölder continuity condition:where and are constants. The term is called the -Laplacian and denoted by .
The study of partial differential equations involving variable-exponent nonlinearities has attracted the attention of researchers in recent years. The interest in studying such problems is stimulated and motivated by their applications in elastic mechanics, fluid dynamics, nonlinear elasticity, electrorheological fluids, and so forth. In particular, parabolic equations involving the -Laplacian are related to the field of image restoration and electrorheological fluids which are characterized by their ability to change the mechanical properties under the influence of the exterior electromagnetic field. The rigorous study of these physical problems has been facilitated by the development of the Lebesgue and Sobolev spaces with variable exponents.
Regarding parabolic problems with nonlinearities of variable-exponent type, many works have appeared. We note here that most of the results deal with blow-up and global nonexistence. Let us mention some of these works. For instance, Alaoui et al. [1] considered the following nonlinear heat equation:in a bounded domain in () with a smooth boundary . Under appropriate conditions on the exponent functions and for , they showed that any solution with nontrivial initial datum blows up in finite time. They also gave a two-dimensional numerical example to illustrate their result. Pinasco [2] studied the following problem:where is a bounded domain with a smooth boundary , and the source term is of the following form:with : and the continuous function : being given functions satisfying specific conditions. They established the local existence of positive solutions and proved that solutions with initial data sufficiently large blow up in finite time. Parabolic problems with sources like the ones in (5) appear in several branches of applied mathematics and they have been used to model chemical reactions, heat transfer, or population dynamics.
Recently, Shangerganesh et al. [3] studied the following fourth-order degenerate parabolic equation:in a bounded domain () with a smooth boundary , and proved the existence and uniqueness of weak solutions of (7) by using the difference and variation methods under suitable assumptions on and the exponents .
Equation is a nonlinear diffusion equation which has been used to study image restoration and electrorheological fluids (see [4–11]). In particular, Bendahmane et al. [12] proved the well-posedness of a solution, for -data. Akagi and Matsuura [13] gave the well-posedness for initial datum and discussed the long-time behaviour of the solution using the subdifferential calculus approach. In our paper, we give an alternative proof of the well-posedness of which is simpler than that in [13] using a theory of nonlinear evolution equations. In addition, we give a numerical example in 2D to illustrate the decay result obtained in [13].
This paper consists of three sections in addition to the introduction. In Section 2, we recall the definitions of the variable-exponent Lebesgue spaces, , the Sobolev spaces, , as well as some of their properties. We also state, without proof, a proposition to be used in the proof of our main result. In Section 3, we state and prove the well-posedness of solution to our problem. In Section 4, we give a numerical verification of the decay result.
2. Preliminaries
We present some preliminary facts about the Lebesgue and Sobolev spaces with variable exponents (see [1, 14–16]). Let be a measurable function, where is a domain of . We define the Lebesgue space with a variable-exponent bywhereis called a modular. Equipped with the Luxembourg-type norm, is a Banach space (see [10]).
Lemma 1 (Hölder’s inequality [10]). Let be measurable functions defined on such that for a.e. . If and , then and
Lemma 2 (see [10]). Let be a measurable function on . Then,(a) if and only if ;(b)for , if , then ; and if , then ;(c).
Lemma 3 (see [10]). If (1) holds, then for any .
We next define the variable-exponent Sobolev space as follows: This space is a Banach space with respect to the norm . Furthermore, is the closure of in . The dual of is defined as , by the same way as the usual Sobolev spaces where .
Lemma 4 (see [10]). Let be a bounded domain of and satisfies (1) and (3), and thenwhere the positive constant depends on and . In particular, the space has an equivalent norm given by .
Lemma 5 (see [10]). If , is a continuous function and Then the embedding is continuous and compact.
Definition 6 (see [17]). Let be a separable Banach space and be a Hilbert space such that with continuous embedding and is dense in . Let be a nonlinear operator.(1) is said to be monotone if If, in addition, we havethen is said to be strictly monotone.(2) is said to be bounded, if is bounded in , whenever is bounded in .(3) is said to be hemicontinuous, if the real functionis continuous from to , for any fixed
We end this section with a proposition which is exactly like Theorem [17].
Proposition 7. Let and . Suppose that : is a bounded monotone and hemicontinuous (nonlinear) operator satisfying, for some and for some ,Then the following problemhas a unique weak solution: where .
3. Well-Posedness
In this section, we state and prove the well-posedness of our problem.
Theorem 8. Let . Assume that (1) and (3) hold. Then has a unique weak solution:where .
Proof. We verify the conditions of Proposition 7. Let , and equip it with the norm, So, . Define byBoundedness of . For all ,Hölder inequality givesCombining (25) and (26), we obtainThen, Lemma 2 impliesCombining (27) and (28), we arrive atLet such that , for all That is, .
If , then Lemma 2 implies and (29) gives .
If , then . Thus, (29) implies .
Hence is bounded.
Monotonicity of . Let Then By using the inequality,for all and a.e. . Thus we obtain .
Hence, is monotone.
To verify (19), we note that, for all , we haveIf , then by Lemma 3, we getCombining (32) and (33), we easily see thatIf , then by Lemma 2, we obtain Therefore, we haveHemicontinuity of . Let be fixed. LetLet (real) and consider Since for a.e. and where , then, by the classical dominated convergence theorem, Hence, is hemicontinuous.
Therefore, conditions of Proposition 7 are satisfied and problem has a unique solution.
4. Numerical Study
In this section, we present some numerical results and applications of the problem:which is a well-posed problem due to Theorem 8. Our objective is to provide a numerical verification of the following decay result:
Proposition 9 (see [13]). Assume that (1) and (3) hold. Then the solution of satisfies the following:(i)If , then there exists a constant such that (ii)If , there exists a constant and such that
We consider two applications to illustrate numerically an exponential decay for the case and a polynomial decay for an exponent function satisfying conditions (1)–(3).
For this purpose, we introduce a numerical scheme for , prove its convergence in Section 4.1, and show the decay results in Section 4.2.
4.1. Numerical Method
In this part, we present a linearized numerical scheme to obtain the numerical results of the system and confirm the decay results. The system is fully discretized through a finite difference method for the time variable and a finite element Galerkin method for the space variable. Useful background about the numerical and error analysis of these methods is found in [18]. More interestingly in [19], Li and Wang introduced a numerical scheme to solve strongly nonlinear parabolic systems and proved unconditional error estimates of the scheme. Our problem is highly nonlinear due to the presence of the gradient and nonlinear exponent in the diffusivity coefficient, which can be zero inside the spatial domain. Below, we introduce our numerical scheme for the purpose of confirming the decay results.
The parabolic equationis discretized using finite differences for the time derivative and a finite element method for the Laplacian term. For this, we divide the time interval into equal subintervals by and denote byThe term is approximated using the first-order forward finite difference formula:Semidiscrete Problem. A linear semidiscrete formulation of takes the following form: given and , find such that This problem is elliptic and admits a unique solution [20], for every . Also, the Rothe approximation to the exact solution , given byis well defined and in as , see [1].
Full-Discrete Problem. The variable is discretized in space by a finite element method. For this, let be a triangulation of with a maximal element size . Let also be a test function in the linear Lagrangian space such that on .
The semidiscrete problem is then written in a weak form to define the full-discrete problem: given , , find such thatFor , the above problem has a unique solution for every nontrivial . This follows from the Lax-Milgram Lemma, and the Galerkin approximation converges to in as ; see [18].
4.2. Numerical Results
In this subsection, we present the following numerical applications of :(1)Exponential decay: for , we show, for some , that(2)Polynomial decay: for , we show, for some and , thatHere, denote the greatest integer function.
In both applications, we set the following parameters:
Figure 1 shows the mesh used for , which involves triangles and vertices.
The initial condition is taken to be and projected into ; see Figure 2.
The numerical results are obtained using the noncommercial software, FreeFem++ [21].
Application 1. satisfies the required conditions (1)–(3). Figure 3 shows the numerical solutions for , , , and .
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(b)
(c)
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With , Figure 4(a) shows that decays exponentially as
(a) Solid line: ; dashed:
(b)
This is also confirmed by Figure 4(b) that shows the ratio is less than one and remains decreasing for a large value of .
Application 2. The exponent function , in Figure 5, satisfies the required conditions (1)–(3) as(i), ;(ii) for with .
Figure 6 shows the numerical solutions for , , , and .
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(b)
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In this case, the solution has a polynomial decay. With , Figure 7(a) shows that This is also confirmed by Figure 7(b), which shows that the ratio remains less than one and decreasing until .
(a) Solid line: ; dashed:
(b)
We conclude that the numerical results in above applications verify Proposition 9.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors acknowledge King Fahd University of Petroleum and Minerals for its support. This work is sponsored by KFUPM under Project no. FT 161004.