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Revisit on global existence of solutions for semilinear damped wave equations in $\mathbb{R}^N$ with noncompactly supported initial data

Yuta Wakasugi

TL;DR

The paper reconsiders global existence for the semilinear damped wave equation in $\mathbb{R}^N$ with noncompact initial data and introduces a polynomial-type weight $\Psi(t,x) = \left(A + \frac{|x|^2}{1+t}\right)^{\lambda}$ to obtain small-data global existence. It proves global mild solutions for exponents $p$ with $p_F(N) < p \le \frac{N}{[N-2]_+}$ under weighted smallness conditions on the initial data, with $\lambda$ chosen above certain lower bounds. The approach relies on a weighted energy method, Matsumura decay estimates, and the Caffarelli–Kohn–Nirenberg inequality to control the nonlinear term, yielding a simpler framework than previous exponential-weight methods. The results extend potential applicability to time-dependent damping and more general domains or potentials through the same weighted-energy strategy.

Abstract

In this note, we study the Cauchy problem of the semilinear damped wave equation and our aim is the small data global existence for noncompactly supported initial data. For this problem, Ikehata and Tanizawa [5] introduced the energy method with the exponential-type weight function $e^{|x|^2/(1+t)}$, which is the so-called Ikehata--Todorova--Yordanov type weight. In this note, we suggest another weight function of the form $(1+|x|^2/(1+t))^λ$, which allows us to treat polynomially decaying initial data and give a simpler proof than the previous studies treating such initial data.

Revisit on global existence of solutions for semilinear damped wave equations in $\mathbb{R}^N$ with noncompactly supported initial data

TL;DR

The paper reconsiders global existence for the semilinear damped wave equation in with noncompact initial data and introduces a polynomial-type weight to obtain small-data global existence. It proves global mild solutions for exponents with under weighted smallness conditions on the initial data, with chosen above certain lower bounds. The approach relies on a weighted energy method, Matsumura decay estimates, and the Caffarelli–Kohn–Nirenberg inequality to control the nonlinear term, yielding a simpler framework than previous exponential-weight methods. The results extend potential applicability to time-dependent damping and more general domains or potentials through the same weighted-energy strategy.

Abstract

In this note, we study the Cauchy problem of the semilinear damped wave equation and our aim is the small data global existence for noncompactly supported initial data. For this problem, Ikehata and Tanizawa [5] introduced the energy method with the exponential-type weight function , which is the so-called Ikehata--Todorova--Yordanov type weight. In this note, we suggest another weight function of the form , which allows us to treat polynomially decaying initial data and give a simpler proof than the previous studies treating such initial data.
Paper Structure (5 sections, 9 theorems, 47 equations)

This paper contains 5 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Energy estimate with a polynomial-type weight
  3. Proof of Theorem \ref{['thm:1']}
  4. Preliminaries
  5. Proof of Theorem \ref{['thm:1']}

Key Result

theorem 1

Let $N \ge 1$ and let $p > 1$ satisfy where $p_F(N) = 1+2/N$. Then, there exist constants $\lambda=\lambda(N,p) > 0$ and $\varepsilon = \varepsilon (N, p, \lambda) > 0$ such that if the initial data $(u_0, u_1) \in H^1(\mathbb{R}^N) \times L^2(\mathbb{R}^N)$ satisfy then the Cauchy problem eq:ndw admits the unique global mild solution.

Theorems & Definitions (19)

  • definition 1
  • theorem 1
  • remark 1
  • remark 2
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 9 more