Sign-changing solutions of the nonlinear heat equation with persistent singularities

Abstract

We study the existence of sign-changing solutions to the nonlinear heat equation ∂tu = Δu + |u|αu on ℝN, N ≥ 3, with 2/N−2 <α<α0, where α0=4/N−4+2√N−1 ∈ (2/N−2,4/N−2), which are singular at x = 0 on an interval of time. In particular, for certain μ > 0 that can be arbitrarily large, we prove that for any u0 ∈ Lloc∞(ℝN\{0}) which is bounded at infinity and equals μ|x|−2/α in a neighborhood of 0, there exists a local (in time) solution u of the nonlinear heat equation with initial value u0, which is sign-changing, bounded at infinity and has the singularity β|x|−2/α at the origin in the sense that for t > 0, |x|2/αu(t,x) → β as |x|→ 0, where β=2/α(N−2−2/α). These solutions in general are neither stationary nor self-similar.