Solution of nonlinear pseudoparabolic equations by semidiscretization in time

Abstract

In the paper existence theorems for nonlinear pseudoparabolic initial-boundary value problems are proved. To this end there is used the method of semidiscretization in time, where the nonlinear time dependent problem is approximated by a sequence of linear elliptic problems. With assumption of local Lipschitz and Holder-Llpschitz conditions with respect to the nonlinearity there is proved the existence of a unique solution in a sufficiently small time interval and convergence of the approximates to the solution. Furthermore, somewhat weakening the assumptions, a compactness criterion yields convergence of a subsequence of approximates and existence of a solution.