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Large time behavior for the classical wave equation with different regular data and its applications

Wenhui Chen, Ryo Ikehata

TL;DR

Extends the classical wave equation analysis by deriving optimal large-time $${L^2}$$-norm behavior under different regular data in $${\mathbb{R}}^n$$. The authors develop a kernel-based Fourier-splitting method, introducing a time-dependent scale $${\mathcal{D}}_{n,\sigma,s}(t)$$ and sharp estimates for the kernel $${\mathcal{I}}_{n,\sigma,s}(t)$$, to obtain precise growth/decay rates across dimension regimes. They identify critical thresholds $$n_0=2s$$ and $$n_1=2+2s$$ governing local stability and large-time stabilization, and show how $${L^{1,\kappa}}$$-type regularity with vanishing moments can yield global stability, while equality cases may cause blow-up. The results are then applied to the wave equation with scale-invariant terms, undamped $$\sigma$$-evolution, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system. These findings illuminate how initial-data regularity controls long-time dynamics and offer tools for related undamped evolution equations.

Abstract

In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-Δu=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial data belonging to $L^2$ or with additional weighted $L^1$ integrabilities. Particularly, some thresholds are discovered for the (local or global in time) stabilization of this quantity. We also apply these results to the wave equation with scale-invariant terms, the undamped $σ$-evolution equation, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system.

Large time behavior for the classical wave equation with different regular data and its applications

TL;DR

Extends the classical wave equation analysis by deriving optimal large-time -norm behavior under different regular data in . The authors develop a kernel-based Fourier-splitting method, introducing a time-dependent scale and sharp estimates for the kernel , to obtain precise growth/decay rates across dimension regimes. They identify critical thresholds and governing local stability and large-time stabilization, and show how -type regularity with vanishing moments can yield global stability, while equality cases may cause blow-up. The results are then applied to the wave equation with scale-invariant terms, undamped -evolution, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system. These findings illuminate how initial-data regularity controls long-time dynamics and offer tools for related undamped evolution equations.

Abstract

In this paper, we mainly consider large time behavior for the classical free wave equation in . We derive some large time optimal estimates for the quantity of solution with initial data belonging to or with additional weighted integrabilities. Particularly, some thresholds are discovered for the (local or global in time) stabilization of this quantity. We also apply these results to the wave equation with scale-invariant terms, the undamped -evolution equation, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system.
Paper Structure (12 sections, 8 theorems, 106 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Estimates of solution in the $L^2$ norm
  5. Some preliminaries on the crucial kernel
  6. Proofs of Theorems \ref{['Thm-L1-data']} and \ref{['Thm-L2-data']}
  7. Proof of Theorem \ref{['Thm-stable']}
  8. Applications of our results to other evolution models
  9. Wave equation with scale-invariant terms
  10. Undamped $\sigma$-evolution equation
  11. Critical Moore-Gibson-Thompson equation
  12. Linearized compressible Euler system

Key Result

Theorem 2.1

Let $0\leqslant s<\frac{n}{2}$. Let us assume $u_0\in L^2$ and $\nabla^s u_1\in L^2\cap L^1$ for the free wave equation Eq-Waves. Then, the solution satisfies the upper bound estimates for large time $t\gg1$. Furthermore, by assuming $u_0\in L^2\cap L^1$ and $\nabla^s u_1\in L^{1,1}$ with $|P_{\nabla^s u_1}|\neq0$, then the solution satisfies the lower bound estimates for large time $t\gg1$. How

Figures (1)

  • Figure 1: Different separating lines in lower dimensions

Theorems & Definitions (21)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 11 more