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

Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B.

Publ. Mat., Barc. 55, No. 1, 185-200 (2011).

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

Keywords:

sign-changing stationary solutions; linearized operator