Università degli Studi di Trieste. Dipartimento di Matematica e Informatica
Abstract
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
