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Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data

Yuta Wakasugi

Abstract

We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the $L^2$-norm of the solution. In Appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data

Abstract

We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the -norm of the solution. In Appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.
Paper Structure (23 sections, 16 theorems, 217 equations, 1 figure)

This paper contains 23 sections, 16 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}: first part
  4. Proof for the compactly supported initial data
  5. Proof for the general case
  6. Proof of Theorem \ref{['thm1']}: second part
  7. Outline of the proof of Proposition \ref{['prop:ex']}
  8. Linear problem
  9. Semilinear problem
  10. Uniqueness of the mild solution
  11. Existence of the mild solution
  12. Blow-up alternative
  13. Continuous dependence on the initial data
  14. Regularity of solution
  15. Approximation of the mild solution by strong solutions
  16. ...and 8 more sections

Key Result

Proposition 1.2

Let $\Omega = \mathbb{R}^n$ with $n \ge 1$, or $\Omega \subset \mathbb{R}^n$ with $n\ge 2$ be an exterior domain with $C^2$-boundary. Let $a(x) \in C(\mathbb{R}^n)$ be nonnegative and bounded. Let and let $(u_0, u_1) \in H^1_0(\Omega) \times L^2(\Omega)$. Then, there exists a unique global mild solution $u$ to dwx. If we further assume $(u_0, u_1) \in (H^2(\Omega) \cap H^1_0(\Omega)) \times H^1_0

Figures (1)

  • Figure 1: Classification of decay rates in $p$ - $\lambda$ plane when $(n,\alpha)=(3,\frac{1}{2})$

Theorems & Definitions (45)

  • Definition 1.1: Mild and strong solutions
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1: SoWa17_AIMSSoWa21_JMSJ
  • Definition 2.2: Kummer's confluent hypergeometric functions
  • ...and 35 more