Blow-up dynamic of solution to the semilinear Moore-Gibson-Thompson equation with memory terms

Abstract

This article is mainly concerned with the formation of singularity for a solution to the Cauchy problem of the semilinear Moore-Gibson-Thompson equation with general initial values and different types of nonlinear memory terms $ N_{\gamma, \, q}(u) $, $ N_{\gamma, \, p}(u_{t}) $, $ N_{\gamma, \, p, \, q}(u, \, u_{t}) $. The proof of the blow-up phenomenon for the solution in the whole space is based on the test function method ($ \psi(x, t) = \varphi_{R}(x)D_{t|T}^{\alpha}(w(t)) $). It is worth pointing out that the Moore-Gibson-Thompson equation with memory terms can be regarded as an approximation of the nonlinear Moore-Gibson-Thompson equation when $ \gamma\rightarrow 1^{-} $. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.

Keywords

• moore-gibson-thompson equation
• general initial values
• nonlinear memory terms
• blow-up
• test function method