On a nonlinear third-order equation

Abstract

A nonclassical nonlinear third-order partial differential equation modeling nonstationary processes in a semiconductor medium is studied. Seven classes of exact solutions of this equation given as explicit, implicit, and quadrature formulas are constructed. It is shown that, among these solutions, there are some bounded globally with respect to time and some approaching infinity on finite time intervals. These results are of interest in view of the following property of initial boundary-value problems for the given equation. Here the method of energy estimates used in many papers to justify the boundedness or unboundedness of solutions does not provide sufficient conditions for the unboundedness of solutions on finite time intervals.