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Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the $L^q$ framework

Wenhui Chen, Junying Gong

Abstract

This manuscript studies some qualitative properties of solutions to the Cauchy problem for the viscous Moore-Gibson-Thompson (MGT) equation. For one thing, by applying the WKB analysis and diagonalization procedure, we derive some $L^p-L^q$ decay estimates and the large time asymptotic profile for a suitable energy term, which provides a new way to treat higher order MGT-type coupled systems. For another, we obtain the global (in time) singular limits in the $L^q$ framework and the higher order asymptotic profile with respect to small thermal relaxation via the multi-scale analysis and the Fourier analysis. Especially, provided the incompatible initial condition between the viscous MGT equation and the strongly damped wave equation, the formation of initial layer is rigorously justified.

Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the $L^q$ framework

Abstract

This manuscript studies some qualitative properties of solutions to the Cauchy problem for the viscous Moore-Gibson-Thompson (MGT) equation. For one thing, by applying the WKB analysis and diagonalization procedure, we derive some decay estimates and the large time asymptotic profile for a suitable energy term, which provides a new way to treat higher order MGT-type coupled systems. For another, we obtain the global (in time) singular limits in the framework and the higher order asymptotic profile with respect to small thermal relaxation via the multi-scale analysis and the Fourier analysis. Especially, provided the incompatible initial condition between the viscous MGT equation and the strongly damped wave equation, the formation of initial layer is rigorously justified.

Paper Structure

This paper contains 17 sections, 7 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Large time profile in the $L^q$ framework
  4. Small thermal relaxation profile in the $L^q$ framework
  5. Large time decay behavior
  6. The first order (in time) coupled system
  7. Subclass I: Diagonalization procedure for small frequencies
  8. Subclass II: Diagonalization procedure for large frequencies
  9. Subclass III: Exponential stability for bounded frequencies
  10. Decay property and large time profile: Proof of Theorem \ref{['Thm-Lp-Lq']}
  11. Singular limits and initial layer
  12. Formal expansions with small thermal relaxation
  13. Formal derivations of small thermal relaxation profiles
  14. The inhomogeneous viscous MGT equation for the error term
  15. Some estimates for the source term in the Fourier space
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Suppose that the initial data $\Psi_0\in(H^{s+M_{p,q,n}}_p)^3$ with $s\geqslant 0$, $1\leqslant p\leqslant 2\leqslant q\leqslant+\infty$, and Then, there is a unique determined Sobolev solution in the sense of the suitable energy term to the viscous MGT equation Eq_MGT such that It satisfies the $L^p-L^q$ decay estimates as well as the refined estimates in which $\Phi$ is the solution to the r

Figures (1)

  • Figure 1: A fast change as $\tau\downarrow0$ for a large fixed time $t=t_0\gg1$

Theorems & Definitions (20)

  • Theorem \oldthetheorem
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem \oldthetheorem
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 10 more