| Partial differential equations (PDE) of parabolic type are encountered in a variety of problems in mathematics, physics, biology, and many other scientific subjects in which irreversible processes can be adequately described by mathematical models. Nonlinear systems of PDE of parabolic type in divergence form are studied in this thesis. New methods to obtain the dissipativity of the associated semiflows are presented. The results are applied to show the existence of the attractors which carry information on the long time behavior of the solutions.; Sufficient conditions for the existence of steady states to semilinear systems, which represent a large class of models in biology and ecology, are derived.; New techniques have been developed to give simpler proofs for Holder regularity of solutions to nondegenerate and degenerate parabolic equations. The results also assert that the solutions converge to the global attractors, whose existence have been established before, in stronger metrics.; A new theory of exponential attractors in Banach spaces is also presented. This result extends the theory of Eden, Foias, Nicolaenko and Temam for the Hilbert spaces. |
