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

摘要

In this paper we consider the following Cauchy problem for the semi-linearwave equation with scale-invariant dissipation and mass and powernon-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Deltau+\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 aglobal (in time) existence result for \eqref{CP abstract} under suitableassumptions on the coefficients $\mu_1, \mu_2^2$ of the damping and the massterm and on the exponent $p$, assuming the smallness of data in exponentiallyweighted energy spaces. On the other hand a blow-up result for \eqref{CPabstract} is proved for values of $p$ below a certain threshold, provided thatthe data satisfy some integral sign conditions. Combining these results we findthe critical exponent for \eqref{CP abstract} in all space dimensions undercertain assumptions on $\mu_1$ and $\mu_2^2$. Moreover, since the globalexistence result is based on a contradiction argument, it will be shown firstlya local (in time) existence result.