The Airy equations with impulsive effect: multi-valued nonlinear perturbations

Abstract

We study the topological regularity of solutions to the Cauchy problem of a (spatial) third-order partial differential equation with a multi-valued perturbation and an impulsive effect. In the framework of the functional space, the principal part of the differential operator corresponds to an Airy operator generating a noncompact $C_0$-group of unitary operators. Our attention is concerned with the $R_\delta$-structure of the solution set for the Cauchy problem. Geometric aspects of the corresponding solution map are also considered. In our main results, no any compactness condition on the impulsive functions is needed. Moreover, we give illustrating examples for the nonlinearity and impulsive functions.