非线性强阻尼波动方程吸引子的正则性及近似惯性流形

Regularity of the Attractor and Approximate Inertial Manifold for Strongly Damped Nonlinear Wave Equations

研究了非线性强阻尼波动方程utt=αuxxt+σ(ux) x-f(u) +g(x)的初边值问题 ,利用线性主算子在相空间生成的解析半群的性质 ,证明了解的光滑性 ,得到了吸引子的正则性 ,构造了近似惯性流形 ,并证明了该方程的任意解轨道在长时间后进入该流形的小邻域中 .

In this paper, the initial boundary value problem of the strongly damped nonlinear wave equation u tt =αu xxt +σ(u x) x-f(u)+g(x) is considered. By using the analylic property of the semigroup generated by the principal operator of the equation in the phase space, the authors prove the smoothing property of the solution, the regularity of the attractor is obtained and approximate inertial manifolds of the equation are constructed. It is proved that arbitrary trajectory of such equation goes into a small neighbourhood of such manifold after large time.