Blow-up of solutions to the one-dimensional semilinear wave equationwith damping depending on time and space variables

Yuta Wakasugi

doi:10.3934/dcds.2014.34.3831

Article Contents

2014, Volume 34, Issue 9: 3831-3846. Doi: 10.3934/dcds.2014.34.3831

This issue Previous Article On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations Next Article Dimension estimates in non-conformal setting

Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

Yuta Wakasugi^1,

1.
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043

Received: June 30, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.

Keywords:

Semilinear damped wave equations,
blow-up.

Mathematics Subject Classification: Primary: 35L71; Secondary: 35B44.

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