Zeitschrift für angewandte Mathematik und Physik

, Volume 63, Issue 1, pp 65–106

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

Article

DOI: 10.1007/s00033-011-0151-2

Cite this article as:
Wu, ST. Z. Angew. Math. Phys. (2012) 63: 65. doi:10.1007/s00033-011-0151-2

Abstract

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form
$$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$
is considered in a bounded domain Ω. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.

Mathematics Subject Classification (2000)

35B37 35B40 35L20 74D05 

Keywords

Global existence Boundary damping General decay Convexity Blow-up Viscoelastic equation 

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.General Education CenterNational Taipei University of TechnologyTaipeiTaiwan

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