In this paper we will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove blow‐up results or global existence (in time) of small data energy solutions.

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.

This paper is a continuation of our recent paper . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L2 scale with additional L1 regularity for the data. In order to deal with this L2 regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.

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In this paper, we study the blow-up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow "wave-like". A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.