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

TL;DR

This paper proves the existence of sign-changing solutions to a nonlinear heat equation with persistent singularities at the origin, which are neither stationary nor self-similar, for a range of exponents.

Contribution

It introduces new solutions with persistent singularities at the origin for the nonlinear heat equation, expanding understanding of singular behaviors in such equations.

Findings

Existence of solutions with prescribed singularity at the origin.

Solutions can be sign-changing and are not necessarily stationary or self-similar.

Solutions exist for arbitrarily large initial singularity magnitude.

Abstract

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = \Delta u + |u|^\alpha u$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < \alpha <\alpha _0$, where $\alpha _0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $\mu >0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $\mu |x|^{- \frac {2} {\alpha }}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $\beta |x|^{- \frac {2} {\alpha }}$ at the origin in the sense that for $t>0$, $ |x|^{\frac {2} {\alpha }} u(t,x) \to \beta $ as $ |x| \to 0$, where $\beta = \frac {2} {\alpha } ( N -2 - \frac {2} {\alpha } ) $. These solutions in general are neither stationary nor self-similar.