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

¹ Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.
² Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944-970 Rio de Janeiro, RJ, Brazil.
³ Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-126, Ciudad de México 01000, Mexico.
⁴ Université Sorbonne Paris Nord, CNRS UMR 7539 LAGA, 99 Avenue J.-B. Clément, 93430 Villetaneuse, France.

^**** Corresponding author: thierry.cazenave@sorbonne-universite.fr

Received: 29 June 2020
Accepted: 18 November 2020

Abstract

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.