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

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.

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Cited by in corpus (8)

• Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case
• Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the $L^q$ framework
• Optimal large-time estimates and singular limits for thermoelastic plate equations with the Fourier law
• Asymptotic behaviors for Blackstock's model of thermoviscous flow
• Asymptotic profiles and singular limits for the viscoelastic damped wave equation with memory of type I
• On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits
• Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow
• $L^p-L^q$ estimates for the dissipative and conservative Moore-Gibson-Thompson equations