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Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

Ryunosuke Kusaba, Tohru Ozawa

TL;DR

We address the large-time behavior of small-amplitude global solutions to the semilinear heat equation $\partial_t u-\Delta u=f(u)$ with supercritical Fujita exponent $p>p_{\mathrm{F}}(n)$. A novel, direct commutator approach between the heat semigroup $e^{t\Delta}$ and algebraic weights is developed to obtain weighted $L^1$-estimates, enabling weighted-initial-data invariance without relying on comparison principles. The paper constructs a nonlinear approximation sequence $u_N$ that captures the asymptotics of $u(t)$ via the linear heat flow $e^{t\Delta}\varphi_1$, and provides explicit higher-order self-similar profiles through Hermite expansions, with remainder bounds of the form $t^{-N\sigma}$ where $\sigma=\frac{n}{2}(p-1)-1>0$. This framework delivers precise, parabolic self-similar asymptotics and offers a robust method potentially applicable to other parabolic or dispersive models.

Abstract

We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.

Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

TL;DR

We address the large-time behavior of small-amplitude global solutions to the semilinear heat equation with supercritical Fujita exponent . A novel, direct commutator approach between the heat semigroup and algebraic weights is developed to obtain weighted -estimates, enabling weighted-initial-data invariance without relying on comparison principles. The paper constructs a nonlinear approximation sequence that captures the asymptotics of via the linear heat flow , and provides explicit higher-order self-similar profiles through Hermite expansions, with remainder bounds of the form where . This framework delivers precise, parabolic self-similar asymptotics and offers a robust method potentially applicable to other parabolic or dispersive models.

Abstract

We present a new method to obtain weighted -estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in , while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.
Paper Structure (6 sections, 15 theorems, 148 equations)

This paper contains 6 sections, 15 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Heat semigroup
  3. Proof of Theorem \ref{['th:P_weight']}
  4. Proofs of Theorems \ref{['th:P_appro']} and \ref{['th:P_aymptotics']}
  5. Proof of Proposition \ref{['pro:P_global']}
  6. Convergence to the asymptotic expansions

Key Result

Proposition 1.1

Let $p>p_{\mathrm{F}} \left( n \right)$. Then, there exists $\varepsilon_{0} >0$ such that for any $\varphi \in \left( L^{1} \cap L^{\infty} \right) \left( \mathbb{R}^{n} \right)$ with $\left\lVert \varphi \right\rVert_{1} + \left\lVert \varphi \right\rVert_{\infty} \leq \varepsilon_{0}$, P has a un

Theorems & Definitions (32)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 22 more