  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 Analysis: Theory, Methods and Applications, 70 (11)
(2009), 3978-3986] is developed. We propose and study a state-dependent
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.
