On blow-up and on global existence of weak solutions to Cauchy problem for some nonlinear equation of the pseudoparabolic type

Abstract

It is a brief exposition of results of the investigation of Cauchy problem for some nonlinear equation of pseudoparabolic type that is a generalisation of some model of semiconductor theory. In the paper, the potential theory for the linear part of the equation is elaborated, which demanded quite intricate technique, which can be used in other equations. The properties of the fundamental solution of this linear part are also of interest, because of the singularity of its 1st time derivative. This is not usual for this type of equations. Also, we obtain sufficiant conditions of solvability and of finite-time blow-up.