Electron. J. Diff. Eqns., Vol. 2006(2006), No. 45, pp. 1-9.

Blow-up of solutions for an integro-differential equation with a nonlinear source

Shun-Tang Wu

Abstract:
We study the nonlinear viscoelastic wave equation
$$ u_{tt}-\Delta u+\int_0^t g(t-s)\Delta u(s)ds=|u|^p u, $$
in a bounded domain, with the initial and Dirichlet boundary conditions. By modifying the method in [15], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.

Submitted February 7, 2006. Published April 6, 2006.
Math Subject Classifications: 35L05, 35L15, 35L70, 37B25.
Key Words: Blow-up; life span; viscoelastic; integro-differential equation.

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Shun-Tang Wu
General Educational Center
China University of Technology
Taipei, 116, Taiwan
email: stwu@cute.edu.tw

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