Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term
© Xu et al.; licensee Springer. 2013
Received: 24 April 2012
Accepted: 11 February 2013
Published: 1 March 2013
In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term . By using the potential well method, we first prove the global existence. Then we prove that when time , the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.
where is the unknown function with respect to the spacial variable and the time variable is a bounded domain.
was suggested and studied by Greenberg et al. [1, 2] from viscoelasticity mechanics in 1968. Under the condition and higher smooth conditions on and the initial data, they obtained the global existence of classical solutions for the initial-boundary value problem of Eq. (1.4).
was studied, and by using the potential well method, the global existence of weak solution was proved under some assumptions on nonlinear functions , , and the initial data. But we do not know how the global solution behaves as the time goes to infinity, namely the asymptotic behaviour of problem (1.1)-(1.3) is still open up to now. In the present paper, we try to study this problem by the multiplier method [17–22].
The main purpose of present paper is to consider the asymptotic behaviour of solution for problem (1.1)-(1.3). Since in the proof of the asymptotic behaviour of solution the global existence theory is required, it is necessary to give the proof of global existence of solution for problem (1.1)-(1.3).
In this paper, we first give some definitions and lemmas (Section 2). Then we prove the global existence of solution (Section 3). Finally, we prove the asymptotic behaviour of solution (Section 4).
In this paper, we denote by , and .
In this section, we will give some definitions and prove some lemmas for problem (1.1)-(1.3).
Remark 2.1 Note that the definitions of and in the present paper are different from those in  and . The definitions given in this paper will be shown more natural and rational because they are a part of the total energy .
is increasing and ;
Proof This lemma follows from and . □
If , then ;
If , then ;
If , then ,
- (i)If , then we have
- (ii)If , then we have
- (iii)If and , then by
which gives (2.1). □
3 Global existence of solution
In this section, we prove the global existence of weak solution for problem (1.1)-(1.3).
Theorem 3.2 Let (H1) and (H2) hold, , . Assume that , . Then problem (1.1)-(1.3) admits a global weak solution and .
On the other hand, since W is an open set in , Eq. (3.2) implies that for sufficiently large n, we have . Next, we prove that for and sufficiently large n. If it is false, then there exists a such that , i.e. and , i.e. . So, by the definition of d, we get , which contradicts (3.6).
Hence there exist u, , η and a subsequence of such that as , in weak-star, and a.e. in , in weak-star and in weakly, in weak-star, , in weak-star, .
On the other hand, from (3.2) and (3.3), we get in , in . Therefore u is a global weak solution of problem (1.1)-(1.3). □
4 Asymptotic behaviour of solution
In this section, we prove the main conclusion of this paper - the asymptotic behaviour of solution for problem (1.1)-(1.3).
- (ii)Furthermore, if and , then for sufficiently large n, there exists a such that(4.2)
which gives (4.4), where , . □
We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11101102), Ph.D. Programs Foundation of the Ministry of Education of China (20102304120022), the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020), the Natural Science Foundation of Heilongjiang Province (A201014), Science and Technology Research Project of the Department of Education of Heilongjiang Province (12521401), Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities.
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