具有非线性记忆项的波动方程的整体吸引子

基于先验估计的方法,在有界开区域Ω∈Rn上证明了具有非线性记忆项的弱阻尼波动方程utt＋αut＋σ｜ut｜mut-Δu-∫0tμ（t-s）｜u（s）｜βu（s）ds＋g（u）=f的整体吸引子的存在性.首先,我们在H0^1（Ω）×L^2（Ω）中建立该方程的解u的一个时间一致先验估计,证明了吸收集的存在性.其次,在空间H0^1（Ω）×L^2（Ω）中,我们把该方程诱导出的半群S（t）分解为S1（t）与S2（t）,然后,我们利用一致能量估计证明了S2（t）的一致衰减性,最后利用格林算子证明了S1（t）的紧性,从而得出S（t）的整体吸引子的存在性.

Global attractors of wave equations with a nonlinear memory term

In this paper, based on a time-uniform priori estimate method, the existence of the global attractor is proved on an open bounded domain Ω∈ R＂for a weakly damped wave equation with a nonlinear memory term utt＋αut＋σ｜ut｜mut-Δu-∫0tμ（t-s）｜u（s）｜βu（s）ds＋g（u）=f We first in the space H0^1（Ω）×L^2（Ω） establish a time-uniform priori estimate of the solution u to the e- quation, and prove the existence of absorbing set. Then we split the semigroup S（t） induced by the a- bove equation into two parts of S1 （t） and S2 （t）. Consequently, we prove the uniform decay of S2（t） by using the uniform energy estimate, and the compactness of S1 （t） by using the Green operator. With these two results we obtain the existence of the global attractor of S（t）.