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Well-posedness and global extensibility criteria for time-fractionally damped Jordan--Moore--Gibson--Thompson equation

Mostafa Meliani, Belkacem Said-Houari

Abstract

In this paper, we consider the Jordan--Moore--Gibson--Thompson with a time-fractional damping term of the type $δ\textup{D}_t^{1-α} Δ\psit$ where we allow the challenging so-called critical case ($δ=0$). This equation arises in the context of acoustic propagation through thermally relaxed media. We tackle the question of long-time existence of the solution. More precisely, the goal of the paper is twofold: First, we establish local well-posedness of the initial boundary value problem, where we also provide a lower bound on the final time of existence as a function of initial data. Second, we prove a regularity result which guarantees, under the hypothesis of boundedness of certain quantities, that the local solution can be extended to be global-in-time.

Well-posedness and global extensibility criteria for time-fractionally damped Jordan--Moore--Gibson--Thompson equation

Abstract

In this paper, we consider the Jordan--Moore--Gibson--Thompson with a time-fractional damping term of the type where we allow the challenging so-called critical case (). This equation arises in the context of acoustic propagation through thermally relaxed media. We tackle the question of long-time existence of the solution. More precisely, the goal of the paper is twofold: First, we establish local well-posedness of the initial boundary value problem, where we also provide a lower bound on the final time of existence as a function of initial data. Second, we prove a regularity result which guarantees, under the hypothesis of boundedness of certain quantities, that the local solution can be extended to be global-in-time.

Paper Structure

This paper contains 13 sections, 8 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Local well-posedness
  4. Conditional global existence
  5. Organization of the paper
  6. Preliminaries
  7. Notation
  8. Inequalities and embedding results
  9. Generalization of the model
  10. Local-in-time well-posedness
  11. Global existence in 2D
  12. Conditional existence in the 3D case
  13. Conclusions and outlook

Key Result

Lemma 2.1

Let $u \in H^2(\Omega)$ with $\Omega$ a domain in $\mathbb{R}^2$, then where $C$ is a constant that depends only on $\Omega$.

Figures (1)

  • Figure 1: Plot of $T$ (red) and $T_0(N_0,T)$ (blue) as a function of $T$.

Theorems & Definitions (18)

  • Lemma 2.1: Lemma 1 in brezis1980nonlinear
  • proof
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Remark 1: Sharpness of the final time of existence estimate
  • Theorem 4.1
  • Corollary 1
  • Remark 2: Comparison with the result of nikolic2024infty
  • proof : Proof of Theorem \ref{['thm:Beale_Kato_Majda_criterion']}
  • ...and 8 more