Study On A Class Of Nonlinear Parabolic Equations With A Nonlocal Diffusion Term

In this thesis,the initial boundary value problem for a class of nonlinear parabolic equations with a nonlocal diffusion term are studied.By means of G-N inequality,Sobolev embedding inequality,potential well theory and Galerkin method,sufficient conditions for the global existence and blow up of weak solution are established.Further more,combining Young's inequality and Bernoulli's inequality,the continuous dependence and exponential growth of the weak solution are studied.In the first chapter,the physical background of reaction-diffusion equation is briefly reviewed,and the known results,research methods and main conclusions are also introduced.In the second chapter,the useful lemma and potential well theory are introduced.In the third chapter,when the growth index-? of diffusion term satisfies ?>p-1/2,the global existence of weak solution is proved by using Galerkin method.Combined with Sobolev embedding inequality and Young's inequality,the continuous dependence of the weak solution is obtained;When the growth index ? of diffusion term satisfies 0<?<p-1/2,with the help of potential well theory,by using Galerkin method,the global existence and continuous dependence of weak solution are proved when the initial function belongs to the stable set.In the fourth chapter,when the growth index ? of diffusion term satisfies 0<?<p-1/2,if the initial function does not belong to the stable set,it is proved that the weak solution will blow up in finite time by using the counter proof method.Combined with Bernoulli's inequality,the exponential growth of the weak solution is also obtained.