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.

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In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type |ut| p in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent p for the nonlinear term satisfies 1 < p (n + 1)/(n − 1) for n 2 and p > 1 for n = 1. In particular, we find the same blow-up range for p as in the corresponding semilinear wave equation with nonlinearity of derivative type.

In this work we determine the critical exponent for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms, when these terms make both equations in some sense "parabolic-like". For the blow-up result the test functions method is applied, while for the global existence (in time) results we use L 2 − L 2 estimates with additional L 1 regularity. (2010). Primary 35L52; Secondary 35B33, 35B44. Mathematics Subject Classification

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: Utt−a2ΔU−b2−a2∇divU+(−Δ)θUt=F(U),(t,x)∈(0,∞)×double-struckR3, where U=Ufalse(t,xfalse)=()Ufalse(1false)false(t,xfalse),Ufalse(2false)false(t,xfalse),Ufalse(3false)false(t,xfalse)normalT with b2 > a2 > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p1,p2,p3 in Ffalse(Ufalse)=()false|Ufalse(3false)|p1,false|Ufalse(1false)|p2,false|Ufalse(2false)|p3normalT to get results for the global (in time) existence of small data solutions.