FINITE-DIMENSIONAL GLOBAL ATTRACTORS FOR PARABOLIC NONLINEAR EQUATIONS WITH STATE-DEPENDENT DELAY

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. This allows us to show that the model considered generates an evolution operator semigroup S-t on a certain space of Lipschitz type functions over delay time interval. The operators S-t are closed for all t >= 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.