Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity
© Gao and Gao; licensee Springer. 2013
Received: 11 March 2013
Accepted: 26 June 2013
Published: 10 July 2013
The aim of this paper is to study the existence and extinction of weak solutions of the initial and boundary value problem for . First, the authors apply the method of parabolic regularization and Galerkin’s method to prove the existence of solutions to the problem mentioned and then obtain the comparison principle by arguing by contradiction. Furthermore, the authors prove that the solution vanishes in finite time and approaches 0 in norm as .
Keywordsnonlinear parabolic equation nonstandard growth condition extinction -Laplace operator
Model (1.1) may describe some properties of image restoration in space and time. Especially when the nonlinear source , the functions , represent a recovering image and its observed noisy image, respectively. In the case when , are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography [1–3]. When p, σ are functions with respect to the space variable and time variable, this problem arises from elastic mechanics, electro-rheological fluids dynamics and image processing, etc.; see [4–9].
To the best of our knowledge, there are only a few works about parabolic equations with variable exponents of nonlinearity. In , Chen, Levine and Rao obtained the existence and uniqueness of weak solutions with the assumption that the exponent , . In , we applied the method of parabolic regularization and Galerkin’s method to prove the existence of weak solutions to problem (1.1) with the assumption that , . In this paper, we generalize the results in . Especially, unlike , we obtain the existence and uniqueness of weak solution not only in the case when , , but also in the case when , . Furthermore, we apply energy estimates and Gronwall’s inequality to obtain the extinction of solutions when the exponents and belong to different intervals; as we know such results are seldom seen for the problems with variable exponents. At the end of this paper, we prove that the solution approaches 0 in norm as by some techniques in convex analysis.
The outline of this paper is the following. In Section 2, we introduce the function spaces of Orlicz-Sobolev type, give the definition of weak solution to the problem and prove the existence of weak solutions with a method of regularization and the uniqueness of solutions by arguing by contradiction. Section 3 is devoted to the proof of the extinction of the solution obtained in Section 2. In Section 4, we get the long time asymptotic behavior of the solution.
2 Existence and uniqueness of weak solutions
and denote by the dual of with respect to the inner product in .
then problem (1.1) has at least one weak solution u satisfying .
The theorems about the uniqueness of weak solutions are as follows.
Then the nonnegative bounded solution of problem (1.1) is unique within the class of all nonnegative bounded weak solutions.
where and .
with a constant independent of ε.
Noticing that , we obtain a contradiction. This means and , a.e. in . □
In the case when , following the lines of the proof of Theorem 2.2, we have the following theorem.
Then the nonnegative solution of problem (1.1) is unique within the class of all nonnegative weak solutions.
3 Localization of weak solutions
In this section, we study the localization of the weak solution to problem (1.1). Namely, we study the extinction of the solution. We discuss the extinction of weak solutions in the case of and , , respectively. Our main results are the following.
where , , are two positive constants.
where , , , are two positive constants.
Remark 3.1 In the case when , , it is not clear whether any bounded nonnegative solution of problem (1.1) vanishes in finite time.
4 Asymptotic behavior of weak solutions
In this section, we study the asymptotic properties of the weak solution to problem (1.1). Namely, we study the long time asymptotic behavior of the solution, our main result is as follows.
Theorem 4.1 Suppose that , . If the following condition is satisfied
where satisfies .
then it is easy to prove that is a convex functional on .
Applying Theorem 5 in , we get that the set is relatively compact in , so .
Step 3. Let . Noting that , then it is easy to prove that is continuous in , so we have .
that is, .
This completes the proof of Theorem 4.1. □
The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.
- Wei DM: Existence, uniqueness and numerical analysis of solutions of a quasilinear parabolic problem. SIAM J. Numer. Anal. 1992, 29(2):484–497. 10.1137/0729029MathSciNetView ArticleGoogle Scholar
- Dibenedetto E: Degenerate Parabolic Equations. Springer, New York; 1993.View ArticleGoogle Scholar
- Zhao JN: On the Cauchy problem and initial traces for the evolution p -Laplacian equations with strongly nonlinear sources. J. Differ. Equ. 1995, 121: 329–383. 10.1006/jdeq.1995.1132View ArticleGoogle Scholar
- Ruzicka M Lecture Notes in Math. 1748. In Electrorheological Fluids: Modelling and Mathematical Theory. Springer, Berlin; 2000.View ArticleGoogle Scholar
- Antontsev SN, Shmarev SI: Anisotropic parabolic equations with variable nonlinearity. Publ. Math. 2009, 53: 355–399.MathSciNetView ArticleGoogle Scholar
- Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 2006, 66: 1383–1406. 10.1137/050624522MathSciNetView ArticleGoogle Scholar
- Acerbi E, Mingione G, Seregin GA: Regularity results for parabolic systems related to a class of non newtonian fluids. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2004, 21: 25–60.MathSciNetGoogle Scholar
- Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 2001, 156(1):121–140.MathSciNetView ArticleGoogle Scholar
- Diening L, Harjulehto P, Hästö P, Rûžička M Lecture Notes in Mathematics 2017. In Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Heidelberg; 2011.View ArticleGoogle Scholar
- Guo B, Gao WJ: Study of weak solutions for parabolic equations with nonstandard growth conditions. J. Math. Anal. Appl. 2011, 374(2):374–384. 10.1016/j.jmaa.2010.09.039MathSciNetView ArticleGoogle Scholar
- Guo B, Gao WJ: Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity. Acta Math. Sci. 2012, 32(3):1053–1062.MathSciNetView ArticleGoogle Scholar
- Simon J:Compact sets in the space . Ann. Math. Pures Appl. 1987, 4(146):65–96.Google Scholar
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