Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms

Huafei  Di; Yadong  Shang; Xiaoxiao  Zheng

doi:10.3934/dcdsb.2016.21.781

Article Contents

2016, Volume 21, Issue 3: 781-801. Doi: 10.3934/dcdsb.2016.21.781

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Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006

Received: March 31, 2015

Revised: September 30, 2015

Published: April 2016

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Abstract

In this paper, we consider the initial boundary value problem for a fourth order pseudo-parabolic equation with memory and source terms. Under suitable assumptions on the function $g$, the initial data and the parameters in the equation, we not only prove the existence of global weak solutions by the combination of the Galerkin method and potential well theory, but also establish an explicit decay rate estimate of the energy adopting the ideas of Marcelo M. Cavalcanti et al. (J. Differ. Equations 203 (2004) 119-158) and Patrick Martinez (ESAIN: Control Optim. Calc. Var. 4 (1999) 419-444). Furthermore, the finite time blow up results for the solutions with both positive and negative initial energy are obtained under certain conditions.

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Mathematics Subject Classification: Primary: 35K35; Secondary: 35A01, 35B44, 35B40.

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