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 .
Keywords:nonlinear 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
We study the existence of weak solutions in this section. Let us introduce the Banach spaces
The theorems about the uniqueness of weak solutions are as follows.
Theorem 2.2Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:
Then the nonnegative bounded solution of problem (1.1) is unique within the class of all nonnegative bounded weak solutions.
Let us consider first the case . By virtue of the first inequality of Lemma 4.4 in , we get
According to the condition (H3), we have
Secondly, we consider the case , . According to the second inequality of Lemma 4.4 in , it is easily seen that the following inequalities hold:
Plugging the above estimates (2.3), (2.4), (2.6) and (2.3), (2.7), (2.9) into (2.2) and dropping the nonnegative terms, we arrive at the inequality
Theorem 2.3Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:
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.
Proof In Definition 2.1, we choose u as a test-function to show
By (3.1), (3.2), we have
By Gronwall’s inequality, we have
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.
Proof Step 1. Let
Similarly, we have
Applying Theorem 5 in , we get that the set is relatively compact in , so .
By the imbedding , Lemmas (2.1)-(2.3) in  and (H4), we have
This completes the proof of Theorem 4.1. □
The authors declare that they have no competing interests.
Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript.
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.
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