Connecting equilibria by blow-up solutions

Pages: 155 - 164, Volume 6, Issue 1, January 2000

doi:10.3934/dcds.2000.6.155       Abstract        Full Text (266.4K)       Related Articles

Marek Fila - Comenius University, 842 15 Bratislava, Slovenia (email)
Hiroshi Matano - Graduate School of Mathematical Sciences, University of Tokyo, Komaba Tokyo, 153-8914, Japan (email)

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.

Keywords:  Semilinear parabolic equation, nonlinear heat equation, blow­up, connecting orbits, zero number.
Mathematics Subject Classification:  35K57, 35B40, 58F39.

Received: November 1999;      Published: December 1999.