Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

TL;DR

This paper demonstrates the non-uniqueness of positive solutions for certain supercritical semilinear heat equations without scale invariance, using monotonicity and self-similar solution transformations.

Contribution

It establishes conditions under which multiple positive solutions exist for supercritical heat equations with specific nonlinearities, extending understanding beyond scale-invariant cases.

Findings

Existence of at least two positive solutions for given initial data

Non-uniqueness linked to the presence of a positive radial singular stationary solution

Construction of solutions via monotonicity and self-similar transformations

Abstract

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}(\mathbb{R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^{\gamma}_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le \gamma<N(p_f-1)/2$, where $p_f:=\lim_{u\to\infty}uf'(u)/f(u)$ is a growth rate of $f$. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of $u(t)$ is based on the monotonicity argument. Transformations of forward self-similar solutions for $f(u)=u^p$ and $e^u$ play a crucial role.