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Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

Masahiro Ikeda, Takahisa Inui, Yuta Wakasugi

TL;DR

The paper addresses the existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data and polynomial nonlinearity. It develops a Besov-space framework, leveraging refined $L^p$-$L^q$ estimates for the linear damped wave propagator and a fractional Leibniz rule to control the nonlinear term $u^p$. The main contributions are local well-posedness for data in suitable homogeneous Besov spaces and global existence for small data when $p \ge 1 + \frac{2r}{n}$, without requiring $L^2$ initial data, thereby extending the global theory to non-$L^2$ data in higher dimensions. This approach relies on paraproduct techniques and the Duhamel formula within Besov spaces to establish contraction properties and rigorous nonlinear estimates.

Abstract

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.

Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

TL;DR

The paper addresses the existence of solutions to the semilinear damped wave equation with non- slowly decaying data and polynomial nonlinearity. It develops a Besov-space framework, leveraging refined - estimates for the linear damped wave propagator and a fractional Leibniz rule to control the nonlinear term . The main contributions are local well-posedness for data in suitable homogeneous Besov spaces and global existence for small data when , without requiring initial data, thereby extending the global theory to non- data in higher dimensions. This approach relies on paraproduct techniques and the Duhamel formula within Besov spaces to establish contraction properties and rigorous nonlinear estimates.

Abstract

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to in general. Our approach is based on the - estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.
Paper Structure (9 sections, 13 theorems, 128 equations)

This paper contains 9 sections, 13 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Basic facts on functions spaces
  4. Preliminaries
  5. Local and global existence
  6. Proof of Proposition \ref{['prop:lplq']}
  7. Proof of the statements in Remark \ref{['rem:frac:Leib']}
  8. Proofs of Lemmas \ref{['lem:meas:psi']} and \ref{['lem:meas:N']}
  9. Proof of Lemma \ref{['lem:XTM']}

Key Result

Theorem 1.1

Let $n \ge 1$ and assume that the nonlinearity $\mathcal{N}(u)$ is given by assum:N. Let $r \in (2, \infty)$ and define $\beta := (n-1) \left(\frac{1}{2} - \frac{1}{r} \right)$. Assume that there exists $s \in \mathbb{R}$ satisfying the following conditions: Let the initial data satisfy where $\chi_{>1}$ is the cut-off function defined in eq:chi later. Then, there exist $T>0$ and a unique soluti

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1: $L^p$-$L^q$ estimates
  • Proposition 2.2: $L^{p}$-$L^{q}$ estimates in Besov norms
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7: Fractional Leibniz rule
  • ...and 15 more