On the critical exponent 'instantaneous blow-up' versus 'local solubility' in the Cauchy problem for a model equation of Sobolev type

摘要

We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form vertical bar del u vertical bar(q). We prove that when q is an element of (1, 3/2] the Cauchy problem in R-3 has no local-in-time weak solution for a large class of initial functions, while when q > 3/2 there is a local weak solution.