Local properties of solutions to non-autonomous parabolic PDEs with state-dependent delays

Abstract

A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and distributed) to be presented simultaneously. The main difficulties arise due to the presence of discrete state-dependent delays since the nonlinear delay term is not Lipschitz on the space of continuous functions. We find conditions for the local existence, uniqueness and study the invariance principle.

17 pages