一类广义抛物型方程组的周期边界问题

PERIOD BOUNDARY PROBLEM OF GENERALIZED PARABOLIC EQUATIONS

本文用 Galerkin 方法讨论非线性抛物型方程组u_t+Au_(xxx)-Bu_(xx)-(gradg(u))_(xx)=f(x,t,u,u_x)(1)具有周期边界条件 u(x+2D,t)=u(x,t),t≥0,x∈R (2)及初始条件 u(x,o)=φ(x),x∈R (3)的整体广义解与整体古典解的存在唯一性。

In this paper we consider period boundary prabem of generalized parabolic equations by Galerkin method for following types:u_t+Au_(xxxx)-Bu_(xx)-(gradg(u))_(xx)=f(x,t,u,u_x)o;(1)u(x+2D,t)=u(x,t),x_eR,t≥o;(2) u(x,o) =(x),w_eR.(3) We have following results:If k≥2+1 then problem(1) ,(2) ,(3) there are unigue global generalized solution u(x,t).If k≥4+1 then problem(1) ,(2) ,(3) there are unigue global classical solution u(x,t).They have continuous derivative D_x~α(u)(o≤α≤k+1) ,D_x~αD_tu(o≤α≤k-1) and generalized derivative D_x~αu(o≤α≤k+2) , D_x~αDu(o≤α≤k),D_x~αD_t^2u(o≤α≤k-4) .