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January 2011, 10(1): 161-177. doi: 10.3934/cpaa.2011.10.161
Blowing up at zero points of potential for an initial boundary value problem
Jong-Shenq Guo 1, and Masahiko Shimojo 2,
Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137, Taiwan
National Center for Theoretical Sciences, Taipei Office, 1, S-4, Roosevelt Road, Taipei 10617, Taiwan
Received January 2010 Revised May 2010 Published November 2010
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.
Keywords: potential, semilinear heat equation, Blow up, threshold solution..
Mathematics Subject Classification: Primary: 35K55; Secondary: 35K2.
Citation: Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
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Jong-Shenq Guo Masahiko Shimojo