带有临界型非线性项的强阻尼波动方程的整体吸引子

Global Attractors of Strongly Damped Wave Equations with Critical Nonlinearities

本文研究一类带有临界型非线性项的强阻尼波动方程.当指数1/2<θ<1时,利用能量泛函的性质,我们证明了由方程导出的C_0半群T(t)的紧性和耗散性,以及整体吸引子的存在性.当θ=1时,利用磨光与逼近,我们研究了磨光半群T_v(t)随t→∞时的一致渐近行为,以及它们在任意有界区间上强收敛到T(t)的一致性,并把T(t)的整体吸引子表示为磨光半群T_v(t)整体吸引子的上半极限.

This paper deals with a class of strongly damped wave equations u tt＋η（- △） θut＋ （-△）u = f（u） with critical nonlinearities. The main task is to prove the existence of the global attractor of C0-semigroup T（t） derived by the wave equation for critical growth indicator ρ = （N ＋ 2）/（N - 2） under Lipshitz and dissipative conditions. In case 1/2 〈 θ ≤ 1, by studying the energy functional attached to T（t）, we prove that, every bounded subset of the energy space is absorbed uniformly by a bounded set B0 independent of the index θ, which combined with the compactness of T（t）, leads to the existence of the global attractor. And in case θ = 1, the method of modification and approximation are adopted. We show that all the modified semigroups Tv（t）（v ∈ （0, 1]） exhibit the same asymptotic behavior as t → ∞, and they converge to T（t） in strong topology uniformly on bounded intervals as v →∞. Based on these properties, we prove the existence of the global attractor, which can be represented by the upper limit of attractors of modified semigroups.