Blow-up for semilinear wave equations with a data of the critical decay having a small loss

It is known that we have a global existence for wave equations with super-critical nonlinearities when the data has a critical decay of powers. In this paper, we will see that a blow-up result can be established if the data decays like the critical power with a small loss such as any logarithmic power. This means that there is no relation between the critical decay of the initial data and the integrability of the weight, while the critical power of the nonlinearity is closely related to the integrability. The critical decay of the initial data is determined only by scaling invariance of the equation. We also discuss a nonexistence of local in time solutions for the initial data increasing at infinity.