Well-posedness Of Solutions To A Class Of Nonlinear Divergence Type Diffusion Equations

The study of nonlinear divergence type diffusion equation is an important sub-ject in the field of partial differential equation.On the one hand,the problems that studied in nonlinear divergence type diffusion equation are from the mathematical model of the field of physics,chemistry and biology and so on,and the study of nonlinear divergence type diffusion equation is of great theoretical significance and practical backgrounds.On the other hand,the study of nonlinear divergence type diffusion equation puts forward a number of challenging issues to the mathemati-cians.So in resent 20 years,an increasing number of mathematicians,physicists,biologists and chemists are interested in this topic and they study it deeply.This thesis is mainly study the weak solutions and entropy solutions of a class of nonlinear divergence type diffusion equation.In this thesis,we recall the develop-ment of this type of equations.Then,by calculus of variations and the approxima-tion argument,we prove the existence and uniqueness of weak solution and entropy solution of this kind of nonlinear divergence type diffusion.The main results are presented as follows:Firstly,we investigate the following nonlinear divergence type diffusion equation with Neumann boundary valuewhere ? is a bounded,open domain of RN(N? 2)with Lipschitz boundary(?)?,n is the outward ulit normal vector of(?)?,T is a positive number and u0?L2(?)The function a is such that ?:R?R defined byis an odd increasing homeomorphism from R onto itself.By assuming the following condition,we prove the existence and uniqueness of weak solutions of this problem.Assume that there exist l,m>1 such thatSecondly,we study the following nonlineaxr divergence type diffusion equation with Dichlet boundary valuewhere ? is a bounded,open domain of RN(N>2)with Lipschitz boundary(?)?,T is a positive number and u0? L2(?).The function a is such that ? R?R defined byis an odd increasing homeomorphism from R onto itself.C1(?)is not dense in W01,1(?),so the existence and uniqueness of weak solutions of this problems are obtained by new structured text function.Thirdiy,suppose ?(?)RN(N>2)is a bounded,open Lipschitz domain,T is a positive number and Q =?×(0,T],? =(?)?×(0,T].We study the well-posedness of entropy solutions for the following nonlinear parabolic equationswhituo u0 L1(?)and f in L1(Q),where the function a is such that ? R?R define byis an odd increasing homeomorphism from R onto itself.Applying the calculus of variations and the approximation argument,we prove the existence and uniqueness of entropy solutions of this equation.