Spectral properties of stationary solutions of the nonlinear heat equation

Abstract

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.