The Research On The Regularity Of Weak Solution Of A Class Of Nonlinear Parabolic Equations

Degenerate parabolic equations are an important class of nonlinear parabolic equations.On one hand,the study of degenerate parabolic equations is of great theoretical significance and practical backgrounds,and it comes from mathematical models of physics,chemistry and biology and other fields.On the other hand,some scholars find several new ideas and methods while overcome the difficulties of degenerate,and promote the development of mathematics.The problems concerning variable exponents in recent years have attracted a considerable amount of attention, and it can be regarded as a class of nonstandard growth problems.It has important applications in nonlimear elasticity, rheological and other physical phenomena.This thesis mainly study the regularity of weak solutions of nonlinear degenerate parabolic equations with p ( x )-growth conditions in the theoretical framework of variable exponents spaces L^p(x)(Ω)and W^1,xL^p(x)(Q ).The method we use is intrinsic scaling,which base on the iterative method by De Giorgi and Moser.This paper consists of the following two parts:The first part , we discuss the regularity of weak solutions in space C (0, T ; L²(Ω)).The weak solutions of time variable in the space W^1,xL^p(x) (Q ) doesn't have any regularity,so we first research the regularity of weak solutions of time variable.Using the convex combination of weak solutions in the space C (0, T ; L²(Ω)),we find a Cauchy sequence which converges to weak solutions.So we prove that weak solutions belong to the space C (0, T ; L²(Ω)).The second part,we investigate the H(o|¨)lder continuity of weak solutions.On the basis of the first part,we construct a sequences of cylinders,and analysis the essence oscillation of the weak solutions,and we get a new cylinder,where the weak solution is H(o|¨)lder continuous,then we use the finite covering theorem to prove the local H(o|¨)lder continuity of the weak solutions.