Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two

TL;DR

This paper investigates sign-changing stationary solutions of the nonlinear heat equation in two dimensions, showing that small perturbations of these solutions lead to finite-time blowup for large p, through spectral analysis of the linearized operator.

Contribution

It provides a detailed analysis of the asymptotic behavior of sign-changing stationary solutions and their linearized operators, establishing conditions for blowup in the nonlinear heat equation.

Findings

Solutions blow up in finite time for small perturbations when p is large.

Eigenvalues of the linearized operator determine stability and blowup behavior.

Asymptotic analysis of solutions and eigenfunctions guides the blowup criteria.

Abstract

Consider the nonlinear heat equation v_t-\Delta v=|v|^{p-1}v in the unit ball of R^2, with Dirichlet boundary condition. Let u_{p,K} be a radially symmetric, sign-changing stationary solution having a fixed number K of nodal regions. We prove that the solution of the equation with initial value \lambda u_{p,K} blows up in finite time if |\lambda-1|>0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of $u_{p,K}$ and of the linearized operator L= -\Delta - p |u_{p,K}|^{p-1}. To show this we consider the linearized operator L= -\Delta - p|u_p|^{p-1} and study the behavior of its first eigenvalue and of its first normalized eigenfunction for large p.