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

We study the existence of sign-changing solutions to the nonlinear heat equation ∂_tu = Δu + |u|^αu on ℝ^N, 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 ∈ L_loc^∞(ℝ^N\{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. [ABSTRACT FROM AUTHOR]