SPECTRAL PROPERTIES OF STATIONARY SOLUTIONS OF THE NONLINEAR HEAT EQUATION

• Published in: Publicacions matemàtiques Vol. 55; no. 1; pp. 185 - 200
• Main Authors: Cazenave, Thierry, Dickstein, Flávio, Weissler, Fred B.
• Format: Journal Article
• Language: English
• Published: Universitat Autònoma de Barcelona 01-01-2011; Universitat Autònoma de Barcelona, Departament de Matemàtiques
• Subjects: 35B35; 35J60; 35K55; Boundary conditions; Eigenfunctions; Eigenvalues; Eigenvectors; finite-time blowup; Heat equation; Integers; linearized operator; Semilinear heat equation; sign-changing stationary solutions; Unit ball
• ISSN: 0214-1493; 2014-4350
• DOI: 10.5565/PUBLMAT_55111_09

In this paper, we prove that if Ψ is a radially symmetric, signchanging stationary solution of the nonlinear heat equation (NLH) ut – Δu = |u|αu, in the unit ball of ℝN, N = 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value λΨ blows up in finite time if |λ — 1| > 0 is sufficiently small and if α > 0 is sufficiently small. The proof depends on showing that the inner product of Ψ with the first eigenfunction of the linearized operator L = -Δ-(α + 1)|Ψ|α is nonzero.