The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case

TL;DR

This paper analyzes the Cauchy problem for the dissipative Moore-Gibson-Thompson equation, deriving improved solution estimates, asymptotic profiles, and studying both existence and nonexistence of solutions in various settings.

Contribution

It provides new $L^2$ estimates, asymptotic profiles, and singular limit results for the dissipative MGT equation, advancing understanding of its linear and semilinear models.

Findings

Derived improved $L^2$ estimates for solutions.

Established asymptotic profiles and approximate relations.

Proved global existence and nonexistence results under certain conditions.

Abstract

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [46]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on these sharp $L^2$ estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.