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Asymptotic behaviour of solution for multidimensional viscoelasticity equation with nonlinear source term

Runzhang Xu1*, Jie Liu1, Yi Niu2 and Shaohua Chen3

Author affiliations

1 College of Science, Harbin Engineering University, Harbin, 150001, People’s Republic of China

2 College of Automation, Harbin Engineering University, Harbin, 150001, People’s Republic of China

3 Department of Mathematics, Cape Breton University, Sydney, B1P 6L2, Canada

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Citation and License

Boundary Value Problems 2013, 2013:42  doi:10.1186/1687-2770-2013-42

Published: 1 March 2013

Abstract

In this paper we study the initial-boundary value problem of the multidimensional viscoelasticity equation with nonlinear source term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M1">View MathML</a>. By using the potential well method, we first prove the global existence. Then we prove that when time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/42/mathml/M2">View MathML</a>, the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.