Existence and non-existence of global solutions for a discrete semilinear heat equation

Keisuke Matsuya; Tetsuji Tokihiro

doi:10.3934/dcds.2011.31.209

Article Contents

2011, Volume 31, Issue 1: 209-220. Doi: 10.3934/dcds.2011.31.209

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Existence and non-existence of global solutions for a discrete semilinear heat equation

Keisuke Matsuya^1, and
Tetsuji Tokihiro^1,

1.
University of Tokyo, Komaba 3-8-1, Meguro, Tokyo 153-8914

Received: March 31, 2010

Revised: September 30, 2010

Published: December 2010

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Abstract

Existence of global solutions to initial value problems for a discrete analogue of a $d$-dimensional semilinear heat equation is investigated. We prove that a parameter $\alpha$ in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for $0<\alpha \le 2/d$, and, for $\alpha > 2/d$, existence of non-trivial global solutions for sufficiently small initial data.

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Mathematics Subject Classification: Primary: 39A14, 74G25; Secondary:35K58, 39A12.

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