Chunyou Sun

Homogenization for nonlinear Schrödinger equations with periodic nonlinearity and dissipation in fractional order spaces

Abstract We study the nonlinear Schrodinger equation with time-oscillating nonlinearity and dissipation originated from the recent studies of Bose-Einstein condensates and optical systems which reads i ψ t + Δ ψ + φ ( ω t ) | ψ | α ψ + i ς ( ω t ) ψ = 0. Under some conditions, we show that as ω → ∞ , the solution ψ ω will locally converge to the solution of the averaged equation i ψ t + Δ ψ + φ 0 | ψ | α ψ + i ζ 0 ψ = 0 with the same initial condition in L q ( ( 0 , T ) , B r , 2 s ) for all admissible pairs ( q , r ) , where T ∈ ( 0 , T max ) . We also show that if the dissipation coefficient ζ o large enough, then, ψ ω is global if ω is sufficiently large and ψ ω converges to ψ in L q ( ( 0 , ∞ ) , B r , 2 s ) , for all admissible pairs (q,r) In particular, our results hold for both subcritical and critical nonlinearities.

Uniform quasi-differentiability of semigroup to nonlinear reaction-diffusion equations with supercritical exponent

A new approach is established to show that the semigroup {S(t)}t≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in Lq (Ω) (2 ≤ q < ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.