On Blow-up and Global Existence of Weak Solutions to Cauchy Problem for Some Nonlinear Equation of the Pseudoparabolic Type

Abstract

We briefly present the results of the investigation of the Cauchy problem for a nonlinear pseudoparabolic equation that is a mathematical generalisation of a certain model in semiconductor theory. The potential theory for the linear part of the equation is elaborated, which demands quite laborious technique, which can be applied for other equations. The properties of the fundamental solution of this linear part are also of interest because its 1st time derivative possesses a singularity. This is not usual for equations of the considered type. Moreover, sufficient conditions for global-in-time solvability are obtained in the paper, as well as sufficient conditions for its finite-time blow-up.