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Self-similar solutions of semilinear heat equations with positive speed

Kyeongsu Choi, Jiuzhou Huang

TL;DR

The authors classify smooth self-similar solutions of the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ with positive speed, using similarity variables and a weighted energy framework anchored by the Ornstein–Uhlenbeck operator. Under either a Gaussian-weighted integral condition or a supercritical exponent $p>1+\sqrt{4/3}$, any such self-similar solution in $\mathbb{R}^n$ must have the explicit form $u(x,t)=\kappa (T-t)^{-1/(p-1)}$ with $\kappa=(1/(p-1))^{1/(p-1)}$. The paper then shows that finite-time blow-up on a bounded convex domain with nonnegative initial data yields convergence, after rescaling, to the same constant profile, i.e., asymptotic self-similarity to $\kappa$. A detailed analysis of the linearized operator around self-similar profiles reveals a precise connection between positive speed and linear stability, implying rigidity: nonzero linearly stable self-similar solutions must have positive speed and correspond to the constant profile $\pm\kappa$. These results extend Liouville-type rigidity without symmetry and clarify blow-up structure and stability for the semilinear heat equation.

Abstract

We classify the smooth self-similar solutions of the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a positive constant after rescaling at the blow-up point for all $p>1$.

Self-similar solutions of semilinear heat equations with positive speed

TL;DR

The authors classify smooth self-similar solutions of the semilinear heat equation with positive speed, using similarity variables and a weighted energy framework anchored by the Ornstein–Uhlenbeck operator. Under either a Gaussian-weighted integral condition or a supercritical exponent , any such self-similar solution in must have the explicit form with . The paper then shows that finite-time blow-up on a bounded convex domain with nonnegative initial data yields convergence, after rescaling, to the same constant profile, i.e., asymptotic self-similarity to . A detailed analysis of the linearized operator around self-similar profiles reveals a precise connection between positive speed and linear stability, implying rigidity: nonzero linearly stable self-similar solutions must have positive speed and correspond to the constant profile . These results extend Liouville-type rigidity without symmetry and clarify blow-up structure and stability for the semilinear heat equation.

Abstract

We classify the smooth self-similar solutions of the semilinear heat equation in satisfying an integral condition for all with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with and converges to a positive constant after rescaling at the blow-up point for all .
Paper Structure (5 sections, 17 theorems, 90 equations)

This paper contains 5 sections, 17 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Self-similar solutions of positive speed
  3. Integral estimates
  4. Proof of and Theorem \ref{['main theo']} and Theorem \ref{['cor']}
  5. Positive speed and linearly stability of self-similar solutions

Key Result

Theorem 1.1

Suppose that $u$ is a smooth self-similar solution of (eq0) on $\mathbb{R}^n\times(0,T)$ about $(a,T)$ with $p>1$, satisfying $u_t(x,0)>0$ for $x\in\mathbb R^n$, and one of the following conditions (1) $\int_{\mathbb{R}^n}|u(x,t)|^{2p}(T-t)^{\frac{2p}{p-1}}e^{-\frac{|x-a|^2}{4(T-t)}}dx<\infty, \quad

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Corollary 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 29 more