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: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta u+\dfrac{\mu_1}{1+t} u_t+\dfrac{\mu_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x), \end{cases}\tag{$\star$} \end{align} where $\mu_1, \mu_2^2$ are nonnegative constants and $p>1$. On the one hand we will prove a global (in time) existence result for \eqref{CP abstract} under suitable assumptions on the coefficients $\mu_1, \mu_2^2$ 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 \eqref{CP abstract} 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 \eqref{CP abstract} in all space dimensions under certain assumptions on $\mu_1$ and $\mu_2^2$. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result.