Spectral properties of stationary solutions of the nonlinear heat equation.(English)Zbl 1211.35151

Summary: We prove that if $$\Psi$$ is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation
$u_t-\Delta u= |u|^\alpha u,$
in the unit ball of $$\mathbb R^N$$, $$N=3$$, with Dirichlet boundary conditions, then the solution 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.

MSC:

 35K58 Semilinear parabolic equations 35B35 Stability in context of PDEs 35J60 Nonlinear elliptic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35B44 Blow-up in context of PDEs
Full Text: