Global Attractor for Nonlinear Wave Equations with Critical Exponent on Unbounded Domain

Asymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearityλup-1uof orderpn<p≤1+2/nfor arbitrarily large initial data, where the lower boundpnis a positive root ofn+2p2-6p-n=0forn≥2andp1=1+2forn=1.Our purpose is to extend the previous results for higher space dimensions concerningL2-time decay and to improve the lower bound ofpunder the same dissipative condition onλ∈C:Im⁡ λ<0andIm⁡ λ>p-1/2pRe λas in the previous works.

Delay-dependent dissipative control for stochastic singular systems with state delay

The problem of delay-dependent dissipative analysis and control for stochastic singular systems with state delay is investigated in this paper. Delay-dependent dissipative condition for the stochastic singular systems with state delay is obtained by employing singular stochastic Lyapunov and LMI-based methods. Based on this condition, a delay-dependent dissipative controller is presented. A numerical example is provided to demonstrate the effectiveness of the proposed approach.