半线性发展方程u_u—2△u_t+△~2u=F(u,u_t,D_xu,D_x^2u)的初值问题

Cauhy Problem for Semilinear Evolution Equation (■_t-Δ)~2u=F(u,u_t,u_x,u_(xx))

本文讨论半线性发展方程(A)(_t-Δ)~2u=F(u_t,D_s^2u)和(B)_t-Δ)~2u=F(u,u_t,D_xu,D_s^2u)的小韧值问题,在对幂性非线性项F■=0(|■^(1+a))之整数a与空间维性n之间满足某些限制条件下,利用线性方程解的L_p-L_q估计和插值定理,设计出基本迭代空间.仿照[1]中对半线性波动方程的处理方法,利用压缩映象原理证明了在时间大范围内解的存在唯一性,也展示了当t→∞时解的某些渐近性质.

The Cauchy problems for semilinear evolution equations (A)_t-Δ)~2u=F(u_z,D_z^2u)and (B)(_1-Δ)~2u=f(u,u_t,D_zu、D_z^2u) with small initial datas are discussed.where F=F=o(||^(1+a)) and integer α≥1.We prove that if the space dimension n and integer α satisfy conditions n/2 α/(1+α)>1,for(A) then the global soutions exist,and some asymptotic behaviors are obtained when t→∞.The methods used here are decay estimates for solutions of linear equations,interpolation theorems and contraction mappings.