Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space

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Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of the solution manifold proposed for ordinary equations in [H.-O. Walther, The solution manifold and C 1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195(1), (2003) 46--65]. The existence of a compact global attractor is proven.