具有分数阶阻尼的波动方程的吸引子的正则性

Regularity of the attractor for semilinear wave equations with fractional damping

研究了具有分数阶衰减项的半线性波动方程解的渐近正则性.当非线性项满足临界增长率时,韩英豪等人证明了上述方程相关联的半群在相空间H_0中具有整体吸引子A.对外力项f和非线性项ф附加一些条件下可以证明上述半群在相空间H_1中具有整体吸引子A_1.显然,A_1■A.如果可以证明A是H_1的有界集,那么由整体吸引子的极大性,反过来的包含关系就成立.但一般情况下,仅仅假设f∈H-1(Ω),不能证明A在H_1中的有界性.对f和ф施加适当的假设条件下证明了A在H_1中的有界性.

In this paper,we investigate the asymptotic regularity of the solutions for the semilinear wave equations with fractional damping.Yinghao Han et al.have proved that for the problem with the critical nonlinearity,the associated semigroup possesses a universal attractor Aon the phasespace H0.With the additional assumptions for the external thermfand nonlinearityф,it is also possible to prove that the semigroup possesses a universal attractor A1 on the phase-space H1.Clearly,A1■A.If we could prove that Ais a bounded subset of H1,then on account of the maximality properties of universal attractors,we would have the reverse inclusion.In general,one cannot have an H1-bound for Aassuming only the external therm f∈H^-1（Ω）.In this paper,we prove the boundedness of Ain the space H1 under certain assumptions for fand ф.