具有周期边界条件的强阻尼波动方程的整体吸引子

Global attractors for strongly damped wave equations with periodic boundary condition

研究了具有周期边界条件的强衰减波动方程解的渐近行为.当非线性项满足临界增长率时(即增长率为5阶时),Carvalho和Cholewa证明了上述方程相关联的半群具有整体吸引子.事实上,他们分析了很广泛的情形,即线性强衰减项为分形线性的情形.此变化是重要的提升.从次临界过渡到临界情况是非常不平凡的,这主要是因为临界情况时嵌入不再是紧的.他们的主要证明技巧是用到了Alekseev的非线性项的常数变易法.笔者用不同的方法,当非线性项比起Carvalho和Cholewa所引进的非线性项更一般的情况下,证明了上述方程整体吸引子的存在性.

In this paper,we investigate the asymptotic behavior of the solutions for the strongly damped wave equations with periodic boundary condition.Carvalho and Cholewa have proved that for the problem with the critical nonlinearity（i.e.,when the growth of nonlinearity is of order 5）,the associated semigroup possesses a universal attractor.Actually,the authors analyze a more general situation with the strong damping term is which of the fractional linear form in place of general linear form.This is a significant progress,since the passage from the subcritical to the critical case is highly nontrivial,mainly due to the fact that in the critical situation the embeddings are no longer compact.The key ingredient of the prove is Alekseev＇s nonlinear variation of constants formula.In this paper,using a different approach,we prove the existence of a universal attractor for problem with a more general nonlinearity than the one used in the paper by Carvalho and Cholewa.