半线性拟双曲方程初边值问题解的长时间性态

Long Time Behaviours of Solutions for Initial Boundary Value Problem of Semilinear Pseudohy- Perbolic Equations

研究半线性拟双曲方程的初边值问题ｕｔ－△ｕｔ＝ｆ（ｕ）（１）ｕ（ｘ，０）＝ｕ０（ｘ），ｕｔ（ｘ，０）＝ｕ１（ｘ）（２）ｕ｜Ω＝０（３）解的长时间性态，用积分估计方法证明了，若ｆ′（ｕ）≤０，ｆ（０）＝０，且ｆ（ｕ）满足一定增长条件，则解ｕ及ｕｔ的Ｈ２模与ｕｔ的Ｌ２模均对０≤ｔ＜∞一致有界。

This paper studies the long time behaviours of solutions for the initial boundary value problem of semilinear pseudohyperbolic equations. u tt -△u t=f(u)(1) u(x,0)=u 0(x),u t(x,0)=u 1(x)(2) u| Ω =0(3) Using integral estimate method,it proves that if f′(u)≤0,f(0)=0 and f(u) satisfy certain growth conditions,then the H 2 norms of solution u and u t and L 2 norm of u tt all are uniformly bounded for 0≤t<∞ ,further more L 2 norm of u t decays exponentially as t→∞ .