Blowing up at zero points of potential for an initial boundary value problem

Jong-Shenq Guo; Masahiko Shimojo

doi:10.3934/cpaa.2011.10.161

Article Contents

2011, Volume 10, Issue 1: 161-177. Doi: 10.3934/cpaa.2011.10.161

This issue Previous Article Traveling waves and their stability in a coupled reaction diffusion system Next Article Boundedness in a class of duffing equations with oscillating potentials via the twist theorem

Blowing up at zero points of potential for an initial boundary value problem

Jong-Shenq Guo^1, and
Masahiko Shimojo^2,

1.
Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137
2.
National Center for Theoretical Sciences, Taipei Office, 1, S-4, Roosevelt Road, Taipei 10617

Received: December 31, 2009

Revised: April 30, 2010

Published: December 2010

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Abstract

We study nonnegative radially symmetric solutions for a semilinear heat equation in a ball with spatially dependent coefficient which vanishes at the origin. Our aim is to construct a solution that blows up at the origin where there is no reaction. For this, we first prove that the blow-up is complete, if the origin is not a blow-up point and if there is no blow-up point on the boundary. Then we prove that a threshold solution exists such that it blows up in finite time incompletely and there is no blow-up point on the boundary. On the other hand, we prove that any zero of nonnegative potential is not a blow-up point for a more general problem under the assumption that the solution is monotone in time.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35K20.

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