The Research About The Weak Solutions Of A Class Of Nonlinear Parabolic Equations

The study of nonlinear parabolic equations is an important subject in the field ofpartial differential equation. On the one hand, the problems that studied in nonlinearparabolic equations are from the mathematical model of the field of physics, chemistryand biology and so on, and the study of nonlinear parabolic equations is of great theo-retical significance and practical backgrounds. On the other hand, the study of nonlinearparabolic equations puts forward a number of challenging issues to the mathematicians.So in resent 20 years, an increasing number of mathematicians, physicists, biologists andchemists are interested in this topic and they study it deeply.This thesis is mainly study the weak solutions of nonlinear parabolic equations withp(x)-growth conditions. In this thesis, we recall the development of this type of equations.Then,by Galerkin approximation and iterative method, we prove the existence, locallyboundedness of weak solutions of this kind of nonlinear parabolic equation. The mainwork of the thesis is summarized as follows.First of all, we introduce the space W1,xLp(x)(Q), which can be considered as aspecial case of the space W1,p(x,t)(Q), where p(x) only depends on the space variablex. We prove the embedding theorem about W1,xLp(x)(Q)→L1(0,T;W01 ,p(x)(?)) andW?1,xLq (x)(Q)→L1(0,T;W?1,q (x)(?)), then we prove that F ? W1,xLp(x)(Q) isrelative compactness in L1(Q), and finally we point out the feasibility and the necessityof studying problem in this space.Secondly, in the space W1,xLp(x)(Q), we study the nonlinear parabolic equation withp(x)-growth conditions, where p(x) only depends on the space variable x. The nonlin-ear part is A(u) = ?diva(x,t,u, u)+a0(x,t,u, u), a(x,t,u, u) and a0(x,t,u, u)satisfy p(x)-growth conditions with respect to u and u, and the right-hand of the equa-tion f(x,t)∈W?1,xLq(x)(Q). We first prove the existence of Galerkin solutions of theequation, then by Galerkin approximation we prove the theorem of the existence of weaksolutions, and finally we give an example to support this theorem.Finally, Based on the existence of weak solutions, we study the local boundednessof the weak solutions. When A satisfies the above structure conditions, we know the weak solutions of the homogeneous equation exist. Then by constructing a convergentsequence, we prove the local boundedness of the weak solutions. At the moment, p(x)satisfies p? > max where p? = inf p(x). we also point out that when p(x)satisfies 1 < p?≤max , the weak solutions of this equation is unboundedness,and we give out an example. Finally, we prove the theorem for a more general case.The study of nonlinear parabolic equations with p(x)-growth conditions in this pa-per are more general than that with the constant p-growth conditions, and it plays moreimportant role in practical application.