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

TL;DR

This paper investigates the well-posedness and conditions for extending solutions globally in time for a time-fractionally damped Jordan--Moore--Gibson--Thompson equation, relevant in acoustic propagation in thermally relaxed media.

Contribution

It establishes local well-posedness, provides a lower bound on existence time, and proves criteria for global extension of solutions in the critical damping case.

Findings

Established local well-posedness of the equation.

Derived a lower bound on the existence time based on initial data.

Proved that solutions can be extended globally if certain quantities remain bounded.

Abstract

In this paper, we consider the Jordan--Moore--Gibson--Thompson with a time-fractional damping term of the type $\delta \textup{D}_t^{1-\alpha} \Delta \psit$ where we allow the challenging so-called critical case ($\delta=0$). 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.