Blow-up for Strauss type wave equation with damping and potential

Abstract

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think of a conjecture that the damping leads to a shift of the critical exponent of the nonlinear term, while the potential does not. We obtain a blow-up result which supports the conjecture, although the existence part is still open. What's more, we give an upper bound of lifespan when the exponent of the nonlinear term is supposed to be critical or sub-critical.