Connecting equilibria by blow-up solutions

Marek Fila; Hiroshi Matano

doi:10.3934/dcds.2000.6.155

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2000, Volume 6, Issue 1: 155-164. Doi: 10.3934/dcds.2000.6.155

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Connecting equilibria by blow-up solutions

Marek Fila^1, and
Hiroshi Matano^2,

1.
Comenius University, 842 15 Bratislava
2.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Tokyo, 153-8914

Received: November 1999

Published: January 2000

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Abstract

We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of $L^1$ connections among equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium $\phi^+$ we mean a function $u$($.,t$) which is a classical solution on the interval $(-\infty,T)$ for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a certain weak sense for $t\in [T,\infty)$ and satisfies $u$($.,t$)$\to \phi^\pm$ as $t\to\pm\infty$ in a suitable sense. The main tool in our analysis is the zero number argument; namely to count the number of intersections between the graph of a given solution and that of various specific solutions.

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Mathematics Subject Classification: 35K57, 35B40, 58F39.

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