Perturbations of self-similar solutions

TL;DR

This paper investigates the nonlinear heat equation, demonstrating the existence of infinitely many solutions near certain singular initial conditions and analyzing conditions under which solutions are nonnegative or sign-changing.

Contribution

It constructs perturbations of self-similar solutions for the nonlinear heat equation in the Sobolev subcritical case, revealing multiple solutions with singular initial data.

Findings

Existence of infinitely many solutions with specific singular initial data.

Nonexistence of nonnegative solutions under certain conditions.

Presence of infinitely many sign-changing solutions.

Abstract

We consider the nonlinear heat equation $u_t = \Delta u + |u|^\alpha u$ with $\alpha >0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) \alpha <4$, for every $\mu \in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = \mu |x-x_0|^{-\frac {2} {\alpha }}$ in a neighborhood of some $x_0\in \Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $\mu |x-x_0|^{-\frac {2} {\alpha }}$ on ${\mathbb R}^N $. Moreover, if $\mu \ge \mu _0$ for a certain $ \mu _0( N, \alpha )\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.