一类多维非线性拟抛物方程的初边值问题

Initial Boundary Value Problem for a Class of Multidimensional Nonlinear Pseudoparabolic Equations

本文将二阶线性椭圆方程的已知结果与压缩映像原理结合研究任意维数的非线性拟抛物方程 a_(ij)(x,t)u_(x_ix_jt)+b_i(x,t)u_(x_it)+c(x,t)u_t=F(x,t,u,Du,D^2u)的初边值问题,在条件(H_1)—(H_4)下,得到了古典解u(x,t)∈C^1([0,T];C^(2+α)())的存在唯一性,讨论了解的光滑性渐适性质与blow-up,推广了很多已知结果。

In this paper,by combining the known results of the linear elliptic equation of second order with the contraction mapping principle,we have studied the initial boundary value problem of the nonlinear pseudoparabolic equations in arbitrary dimensions Under the conditions (H1) - (H4),we obtain the existence and uniqueness of the classical solution u(x,t) C1 ([0,T]iC2+a). Moreover the smoothness,asymptotic behavior and blow-up of solution are discussed,so many known results are generalized.