耦合拟线性抛物型方程组的Dirichlef问题弱解的最大模估计

THE ESTIMATION ON MAXIMUM MODULUS OF WEAK SOLUTION FOR DIRICHLET PROBLEMS OF COUPLED QUASI-LINEAR PARABOLIC SYSTEMS

1.解的最大模估计是椭圆型和抛物型偏微分方程研究中的一个极其重要的步骤。在单个方程的情形,基于极值原理,此问题已得到比较满意的解决。但在方程组的情形,由于极值原理一般说来不再成立,解的最大模估计仍有不少困难。1975年以来,H.F.Weinberger,R.Redheffer,W.Walter 和 K.N.Chueh,C.C.Conley,J.A.Smoller 等用不变集的概念。

In this paper,the author considers the weak solutions for Dirichet prob- lems of coupled quasi-linear parabolic systems where Q_T=Ω×[0,T] with bounded open setΩR^n and T>0,T_T is the pa- rabolic boundary of Q_T,U∈R^N is the solution vector and ▽U is the N×n ma- trix consisting of column vectors ,i=1,2,...,n,F,U_0∈R^N,N×N matrix A and functions a_(ij) are given quantities.Suppose that the matrix[a_(ij)] is def- inite positive and all the left eigenvalues of A(U)are real and have a uni- form positive lower bound.So this is a coupled parabolic system. Using Ladyzenskaja's method,this paper gives a estimate of maximum module for the solution vector U in following form: under some assumptions about the parabolic system and the non-negative smooth function G.