A Note on Lp- Lq Estimates for Semilinear Critical Dissipative Klein–Gordon Equations

In this note, we consider a semilinear wave equation with scale-invariant mass and dissipation. The scale of the dissipation is the same of the mass term and this creates an interplay in which a relation between the coefficients comes into play to determine if the dissipation is dominant with respect to the mass, or viceversa. We prove global existence of small data solutions for supercritical power nonlinearities when the mass term is dominant with respect to the dissipative term (“Klein–Gordon type” case). The critical exponent is a modified Strauss exponent for global small data solutions to semilinear Klein–Gordon equations.