Open Access
2011 Spectral properties of stationary solutions of the nonlinear heat equation
Thierry Cazenave, Flávio Dickstein, Fred B. Weissler
Publ. Mat. 55(1): 185-200 (2011).


In this paper, we prove that if $\Psi $ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

\begin{equation} \label{fAbs} u_t -\Delta u= |u|^\alpha u, \tag{NLH} \end{equation}

in the unit ball of ${\mathbb R}^N $, $N= 3$, with Dirichlet boundary conditions, then the solution of \eqref{fAbs} with initial value $\lambda \Psi$ blows up in finite time if $|\lambda -1|>0$ is sufficiently small and if $\alpha >0$ is sufficiently small. The proof depends on showing that the inner product of $\Psi $ with the first eigenfunction of the linearized operator $L= -\Delta - (\alpha +1) | \Psi | ^\alpha $ is nonzero.


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Thierry Cazenave. Flávio Dickstein. Fred B. Weissler. "Spectral properties of stationary solutions of the nonlinear heat equation." Publ. Mat. 55 (1) 185 - 200, 2011.


Published: 2011
First available in Project Euclid: 25 February 2011

zbMATH: 1211.35151
MathSciNet: MR2779581

Primary: 35B35 , 35J60 , 35K55

Keywords: finite-time blowup , linearized operator , semilinear heat equation , sign-changing stationary solutions

Rights: Copyright © 2011 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.55 • No. 1 • 2011
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