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

TL;DR

This work shows nonuniqueness for the Cauchy problem for the semilinear heat equation $\partial_t u-\Delta u=f(u)$ in $\mathbb{R}^N$ with non-scale-invariant nonlinearities by exploiting the existence of a positive radial singular stationary solution $u^*$. The authors construct a second solution via a monotone iteration built from a carefully crafted supersolution that combines a singular profile with forward self-similar transformations of canonical nonlinearities. In the supercritical regime $q_{JL}<q_f<q_S$ (equivalently $p_S<p_f<p_{JL}$), they prove the existence of two positive solutions with the same initial data $u_0=u^*$; a second solution $u(t)$ stays locally bounded for $0<t<t_0$ and converges to $u^*$ in $L^{\gamma}_{ul}$ as $t\to0^+$ for $1\le \gamma<\gamma^*$. A parallel nonuniqueness result holds in the critical/subcritical range $q_S\le q_f<q_0$ under an additional near-origin profile condition (A8). These results unify and extend known nonuniqueness phenomena for power and exponential nonlinearities and highlight the pivotal role of singular stationary solutions in the loss of uniqueness, beyond scale-invariant settings.

Abstract

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] 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 $$ u(t)\to u^*\quad\text{in}\ L^γ_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le γ<N(p_f-1)/2$, 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.

Theorems & Definitions (25)

• Definition 1.1
• Proposition 1.2
• Lemma 1.3
• Theorem 1.4: Nonuniqueness (supercritical)
• Theorem 1.5: Nonuniqueness (critical/subcritical)
• Remark 1.7
• Lemma 3.1
• proof
• Proposition 3.2
• Lemma 3.3