Rezunenko O. Qualitative properties of dynamical systems generated by nonlinear delay partial differential equations

We develop novel methods to study models of mathematical physics which are described by nonlinear delay differential equations and systems of different types: constant and state-dependent, discrete and distributed, local and nonlocal in space coordinates. Main interest is in the qualitative behaviour of systems concerning Lyapunov stability, existence of Approximate inertial manifolds, Inertial manifolds with delay and global attractors. The central part of our in-vestigations is devoted to state-dependent delay. We present novel methods for proving the well-posed solvability of state-dependent delay partial differential equations. The results are obtained in different directions: metric space approach and solution manifolds approach. An alternative approach is also proposed based on a new idea which is connected to the so-called “ignoring condition” on the state-dependent delay. Obtained results are applied to a biological problem concerning viral in-host dynamics.