  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 there are two
kinds of blow-up phenomenons when the exponent of the nonlinear term is small.
It also means there are two kinds of law to determine the critical exponent. In
this paper, we obtain a blow-up result and get the estimate of the upper bound
of the lifespan in critical and sub-critical cases. All of the results support
such a conjecture, although for now, the existence part is still open.
