  We deal with a class of parabolic nonlinear evolution equations with
state-dependent delay. This class covers several important PDE models arising
in biology. We first prove well-posedness in a certain space of functions which
are Lipschitz in time. We show that the model considered generates an evolution
operator semigroup $S_t$ on a space $CL$ of Lipschitz type functions over delay
time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous
for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses
compact global and exponential attractors of finite fractal dimension. Our
argument is based on the recently developed method of quasi-stability estimates
and involves some extension of the theory of global attractors for the case of
closed evolutions.
