# On the multiplicity of self-similar solutions of the semilinear heat equation

## TL;DR

This paper investigates the number of positive radial self-similar solutions to a semilinear heat equation, revealing a transition from countably infinite to finitely many solutions as the exponent varies between critical thresholds.

## Contribution

It establishes the exact multiplicity of solutions in different parameter regimes, resolving two open questions about solution counts for these equations.

## Key findings

- Countably many solutions for p in (p_S, p_JL)
- Finitely many solutions for p in (p_JL, p_L)
- Answers to open questions on solution multiplicity

## Abstract

In studies of superlinear parabolic equations \begin{equation*} u_t=\Delta u+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) = (T-t)^{-1/(p-1)}w(y)$, where $y:=x/\sqrt{T-t}$, $T$ is a constant, and $w$ is a solution of the equation $\Delta w-y\cdot\nabla w/2 -w/(p-1)+w^p=0$. We consider (classical) positive radial solutions $w$ of this equation. Denoting by $p_S$, $p_{JL}$, $p_L$ the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for $p\in (p_S,p_{JL})$ there are only countably many solutions, and for $p\in (p_{JL},p_L)$ there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.