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

Abstract

In this work, the Cauchy problem for the semilinear Moore -Gibson -Thompson (MGT) equation with power nonlinearity |u| p on the righthand side is studied. Applying L 2 −L 2 estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow -up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills 1 < p p Str (n) for n 2 and p > 1 for n = 1. Here the Strauss exponent p Str (n) is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case p = p Str (n) a different approach with a weighted space average of a local in time solution is considered.

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

“…Our next goal is to derive a sequence of lower bounds for F 1 by using (18). The approach in the iteration argument is similar as the one in [9]. More precisely, we prove that…”

“…Moreover, modifying slightly the proof of Theorem 3.1 in [9] one can prove the existence of local in time energy solutions with support contained in the forward cone…”

“…[39,Theorem 5.1]). Moreover, a blow -up result for the conservative case with power nonlinearity can be found in [9]. Let us provide some results which are related to our model (2).…”

“…In order to prove this result, we are going to apply an iterative argument for a suitable time -dependent functional, which depends on a local (in time) solution to (2). For the choice of the functional we follow [22] whereas concerning the iteration procedure we use some key ideas from [9], where a technique to deal with an unbounded exponential multiplier in the iteration frame is developed. This approach is based on the idea of slicing the interval of integration and it has been introduced by Takamura and coauthors in the study of critical cases for wave models (see [2,43,44,47] for example).…”

Blow‐up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations

“…Upper bound lifespan estimates of solutions to problem () with power nonlinearities , and derivative nonlinearities , are verified in Theorems 1.1 and 1.2. We simplify the proofs of Theorems 1.1 and 1.2 compared with the proofs of the Cauchy problems for single MGT equation in Chen and Palmieri, 4,5 which has been investigated by making use of iteration approach. Namely, it is a special case in problem () with power nonlinearities , and derivative nonlinearities , , when p = q .…”