A note on the Moore–Gibson–Thompson equation with memory of type II

Abstract

We consider the Moore-Gibson-Thompson equation with memory of type IIwhere A is a strictly positive selfadjoint linear operator (bounded or unbounded) and α, β, γ > 0 satisfy the relation γ ≤ αβ. First, we prove a well-posedness result without requiring any restriction on the total mass ̺ of g. Then we show that it is always possible to find memory kernels g, complying with the usual mass restriction ̺ < β, such that the equation admits solutions with energy growing exponentially fast. In particular, this provides the answer to a question raised in [2].2000 Mathematics Subject Classification. 35B35, 35G05, 45D05.

Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case

“…The conservation of the energy stated in Proposition 1 leads immediately to the uniqueness of the solution to the Cauchy problem (5)…”

“…Introduction. In recent years, the Moore -Gibson -Thompson (MGT) equation, a linearization of a model for wave propagation in viscous thermally relaxing fluids, has caught a lot of attention (see [29,43,14,20,19,28,18,25,35,3,6,24,7,23,34,2,5] and references therein). This model is realized through the third order hyperbolic partial differential equation τ u ttt + u tt − c 2 ∆u − b∆u t = 0.…”

“…By applying the same approach used to prove the existence of solutions in the classical energy space to the Cauchy problem for free wave equation (e.g. Chapter 14 in [10]) and the mean value theorem, we may conclude the existence of solutions to the MGT equation (5).…”

“…Let us introduce some notations for the proof of the local (in time) existence of solutions. We denote by K 0 (t, x), K 1 (t, x) and K 2 (t, x) the fundamental solutions to (5) with initial data (u 0 , u 1 , u 2 ) = (δ 0 , 0, 0), (u 0 , u 1 , u 2 ) = (0, δ 0 , 0) and (u 0 , u 1 , u 2 ) = (0, 0, δ 0 ), respectively. Here δ 0 is the Dirac distribution in x = 0 with respect to spatial variables.…”

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

“…Over the last years, the Moore -Gibson -Thompson (MGT) equation (cf. [30,45]), a linearization of a model for the wave propagation in viscous thermally relaxing fluids, has been studied by several authors (see, for example, [13,20,19,28,18,27,7,11,26,12,25,37,3,10,38,5,6,4,8]).…”