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: u_{tt}−Δu+μ_1 (1+t)^{-1}u_t+ μ_2^2 (1+t)^{-2}u=|u|^p, u(0,x)=u_0(x), u_t(0,x)=u_1(x) (⋆), where μ_1,μ_2^2 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 μ_1,μ_2^2 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 μ_1 and μ_2^2. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result.