Local Behavior Of Solutions Of Quasilinear Parabolic Equations

The Local behavior of weak solutions, for example the boundness , Harnack inequality and Holder continuity, is main component of regularity theory of elliptic and parabolic equations. Recently Zamboni has discussed the boundness, Harnack inequality and Holder continuity of non-negative weak solutions to the following quasilinear elliptic equationsdiv A (x, u, (?)u) + B(x, u, (?)u) = 0under the assumptions of the coefficients for the structure conditions being in Morrey spaces or more general spaces. Aronson and Serrin had studied the boundness . Harnack inequality for non-negative weak solutions to the following quasilinear parabolic equationsu_t = divA(x, t, u, (?)u) + B(x, t, u, (?)u).Here the coefficients of the structure conditions belong to Sobolev spaces L^p,q(Q).The aim of this thesis is to extend Zamboni's works to parabolic equations. By defining a new functional space and using classical Moser's iteration, we can obtain the boundness, Harnack inequality and Holder continuity of weak solutions to the above quasilinear parabolic equations.The background and history for the studied problem are given in chapter 1.In chapter 2, we shall give some functional spaces and lemmas.We give the main results and theirs proof in chapter 3-5.