A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (11) (2009) 3978–3986] is developed. We propose and study an analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions C. The dynamical system is constructed in C and the existence of a compact global attractor is proved.