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Asymptotic expansion for a class of second-order evolution equations in the energy space and its applications

Motohiro Sobajima

Abstract

In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form $u''+Au+u'=0$ in real Hilbert spaces, where $A$ is a nonnegative selfadjoint operator. The main result is the asymptotic expansion for all initial data belonging to the energy space, which is naturally expected. This is an improvement for the previous work (required a sufficient regularity). As an application, we focus our attention to damped wave equations in an exterior domain in $\mathbb{R^N (N \geq 2)$ with the Dirichlet boundary condition. By using the asymptotic expansion in the present paper, we could derive the optimal decay rates of energy functional of solutions to damped wave equations. Moreover, some decay estimates of the local energy (energy functional restricted in a compact subset) can be observed via the asymptotic expansion.

Asymptotic expansion for a class of second-order evolution equations in the energy space and its applications

Abstract

In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form in real Hilbert spaces, where is a nonnegative selfadjoint operator. The main result is the asymptotic expansion for all initial data belonging to the energy space, which is naturally expected. This is an improvement for the previous work (required a sufficient regularity). As an application, we focus our attention to damped wave equations in an exterior domain in with the Dirichlet boundary condition. By using the asymptotic expansion in the present paper, we could derive the optimal decay rates of energy functional of solutions to damped wave equations. Moreover, some decay estimates of the local energy (energy functional restricted in a compact subset) can be observed via the asymptotic expansion.

Paper Structure

This paper contains 12 sections, 25 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Analysis for the abstract evolution equation
  3. Basic facts with a derivation
  4. Asymptotic expansion
  5. Asymptotic expansion for regular initial data (revisited)
  6. Asymptotic expansion for general initial data
  7. Application to the damped wave equation
  8. Decay estimates for $L^2$-norm and the energy functional
  9. Optimality of decay rates for $\Omega=\mathbb{R}^N$ $(N\in\mathbb{N})$
  10. Optimality of decay rates for the exterior problem ($N\geq 3$)
  11. Decay estimates for the two-dimensional exterior problem
  12. Estimates for the local energy

Key Result

Theorem 1.1

For every $n\in \mathbb{Z}_{\geq 0}$, there exists a positive constant $C_{1,n}$ (depending only on $n$) such that the following assertion holds: For every $(u_0,u_1)\in \mathcal{H}$, the corresponding weak solution $u$ of intro:abst-dw satisfies where $\{V_n(\cdot)\}_{n\in \mathbb{Z}_{\geq0}}$ is given in Definition def:asymptotics.

Theorems & Definitions (57)

  • Definition 1.1: Definition of weak solutions
  • Definition 1.2: Sobajima_MathAnn
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.2
  • Remark 1.3
  • Corollary 1.3
  • Remark 1.4
  • Proposition 1.4
  • ...and 47 more