一类非线性抛物方程组弱解的处处正则性

Everywhere HO¨lder Contiminty for Weak Solutions of a Class of Nonlinear Parabolic Equation Systems

二次增长的非线性抛物方程弱解的正则性研究已有了比较完备的结果,但对于非线性抛物方程组弱解的正则性研究取得的成果还不多,有关文献证明了对角型抛物方程组的弱解在一定条件下是HO¨lder连续的.本文考虑如下的一类非线性抛物方程组utk-Dα﹂Akα(z,u,Du)」=Bk(z,u,Du),在满足|Aαk(z,u,0,…,0,pk+1,…,pN)|≤C∑Nj=k+1|pj|1-ε0+fkα(z),这里,ε0∈(0,1),k=1,2,…,N,z=(x,t)∈Ω×(o,T)Rn+1,证明了在一定的增长条件下,其弱解是处处Hlder连续的.

It has appeared many results about regularity for weak solutions of quadric jncreasing parabolic equations.However,there are only a few results about regularity for weak solutions of nonlinear parabolic equation systems.It has been proven in literature that weak solutions of a class of nonlinear parabolic equation systems are continuous under the certain condition.Consider a class of nonlinear parabolic equation systems of the form uk^t-Dα﹂Ak^α（z,u,Du）」=Bk（z,u,Du）,satisfying ｜Ak^α（z,u,0,…,0,p^（k＋1）,…,p^N）｜≤C^N∑（j=k＋1）｜p^j｜^（1-ε0）＋f k^α（z）,where ε0∈（0,1）,k=1,2,…,N,z=（x,t）∈Ω×（o,T）R^（n＋1）,Ω is a bounded open set in Rn.It isshown that the weak solutions of the class of nonlinear parabolic equation systems are Hlder continuous under the certain condition,which generalizes the relative results in literature.