On the Blow-up of Finite-Time Solutions and the Global Solvability of the Cauchy Problem for a Nonlinear Equation in Hölder Spaces

We investigate solvability and blow-up of the classical solution of the Cauchy problem for a certain pseudoparabolic equation, which is related to nonstationary processes in semiconductors. The equation that we consider is interesting; in particular, because of the singularity on time (in t=0) of the fundamental solution.We obtain Schauder-type estimates for the potentials, which allow to establish the smoothness of the classical solution of the Cauchy problem. We also investigate the influence of the nonlinearity exponent on the global solvability of the Cauchy problem with various initial data.