A Nonexistence Result for the Semi-Linear Moore-Gibson-Thompson Equation with Nonlinear Memory on the Heisenberg Group

Abstract

The Moore-Gibson-Thompson theory was developed starting from a third order differential equation, built in the context of some consideration related fluid mechanics. Subsequently the equation was considered as a heat conduction equation because it has been obtained by considering a relaxation parameter into the type III heat conduction. Since the advent of the Moore-Gibson-Thompson theory, the number of dedicated studies to this theory has increased considerably. The Moore-Gibson-Thompson equation modifies and defines equations for thermal conduction and mass diffusion that occur in solids. In this paper we investigate a class of Moore-Gibson-Thompson equation with nonlinear memory on the Heisenberg group.The problem of nonexistence of global weak solutions in the Heisenberg group has received specific attention in the recent years. In the present paper we use the method of test functions to prove nonexistence of global weak solutions. The results obtained in this paper extend several contributions and we focus on new nonexistence results which are due to the presence of the fractional Laplacian operator of order $\frac{\sigma}{2}$.