Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces

In this paper we consider the following Cauchy problem for the semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity:(⋆)utt−Δu+μ11+tut+μ22(1+t)2u=|u|p,u(0,x)=u0(x),ut(0,x)=u1(x), where μ1,μ22 are nonnegative constants and p>1. On the one hand we will prove a global (in time) existence result for (⋆) under suitable assumptions on the coefficients μ1,μ22 of the damping and the mass term and on the exponent p, assuming the smallness of data in exponentially weighted energy spaces. On the other hand a blow-up result for (⋆) is proved for values of p below a certain threshold, provided that the data satisfy some integral sign conditions. Combining these results we find the critical exponent for (⋆) in all space dimensions under certain assumptions on μ1 and μ22. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result.