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Large time asymptotic behavior for the weakly damped Jordan-Moore-Gibson-Thompson equation

Wenhui Chen, Yan Liu, Manqing Luo

Abstract

This manuscript considers the Jordan-Moore-Gibson-Thompson (JMGT) equation and its linearized equation with an additional weak damping term (proposed by [B. Kaltenbacher, \emph{Inverse Problems} (2025)] firstly) in the whole space $\mathbb{R}^n$. We mainly study the unique existence and large time behavior, including optimal decay estimates and asymptotic profiles, of global in-time Sobolev solutions for any $n\geqslant 1$. This weak damping term leads to diffusion profiles in the sub-critical case $δ>0$ and regularity-loss decay properties in the critical case $δ=0$, which are greatly different from the results for the corresponding classical models without the weak damping term.

Large time asymptotic behavior for the weakly damped Jordan-Moore-Gibson-Thompson equation

Abstract

This manuscript considers the Jordan-Moore-Gibson-Thompson (JMGT) equation and its linearized equation with an additional weak damping term (proposed by [B. Kaltenbacher, \emph{Inverse Problems} (2025)] firstly) in the whole space . We mainly study the unique existence and large time behavior, including optimal decay estimates and asymptotic profiles, of global in-time Sobolev solutions for any . This weak damping term leads to diffusion profiles in the sub-critical case and regularity-loss decay properties in the critical case , which are greatly different from the results for the corresponding classical models without the weak damping term.

Paper Structure

This paper contains 21 sections, 11 theorems, 110 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Backgrounds for the classical JMGT equation
  3. Main purposes of this manuscript
  4. Notations
  5. Main results
  6. Weakly damped MGT equation
  7. Asymptotic behavior of solutions in the Fourier space
  8. Asymptotic expansions for the characteristic roots
  9. Asymptotic representations and pointwise estimates of solutions
  10. Large time qualitative properties of solutions
  11. The sub-critical case $\delta>0$
  12. The critical case $\delta=0$
  13. A comparison between the sub-critical and critical cases
  14. Weakly damped JMGT equation in the sub-critical case
  15. Preliminary results
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $\tau>0$, $\delta>0$, $0<\gamma\neq\frac{1}{4\tau}$, and $\sigma\in\{j-2,s\}$ for $j\in\{0,1,2\}$. Let $(\psi_0,\psi_1,\psi_2)\in (H^{s+2}\cap L^1)\times (H^{s+1}\cap L^1)\times (H^s\cap L^1)$ with $s>\max\{\frac{n}{2}-1,0\}$ for any $n\geqslant 1$ being sufficiently small in the corresponding t satisfying the optimal decay estimate Moreover, the refined estimate and the optimal lower bound

Theorems & Definitions (25)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • ...and 15 more