On the blow-up of the solution and on the local and global solvability of the Cauchy problem for a nonlinear equation in Hölder spaces

We consider a Sobolev-type equation that describes a transient process in a semiconductor in an external magnetic field. We obtain the following result depending on the power q of the nonlinear term. When q∈(1,3], the Cauchy problem has no local weak solution. For q>3, we prove a theorem on non-extendable solution. In the latter case, the solution exists globally in time for “small” initial data, but it experiences the blow-up in finite time for sufficiently “large” data. As a technique, in particular, we obtain Schauder-type estimates for potentials. We use them to investigate smoothness of the weak solution to the Cauchy problem. © 2021 Elsevier Inc.