A Study On Some Problems Of Non-autonomous Nonlinear Schrodinger Equations

This dissertation devotes to some non-autonomous nonlinear Schrodinger equa-tions which appear in the studies of Bose-Einstein condensates and optical. Because the non-autonomy of equations leads to some essential mathematical difficulties. For example, the energy is no longer conserved, Scaling invariance and Galilean invariance do not hold, and so on. In this thesis, we focus on some properties of solutions, mainly including some local existence, global existence, the existence and concentration prop-erties of blow-up solutions, averaging of time-oscillating equation, the limit behavior, an optimal bilinear control of time-dependent singular potentials.In Chapter1, we introduce some background, methods and well-known results of classical and non-autonomous nonlinear Schrodinger equations. We also give a com-parison between the classical and non-autonomous equations.In Chapter2, we present some preliminaries.In Chapter3, we study the Cauchy problem for the nonlinear Schrodinger equation with time-dependent loss/gain which reads ut+â–³u+λ|u|αu+ia(t)u=0. We obtain some global existence and blow-up results which depend on the size of the loss/gain coefficient. In particular, we prove the global existence for the energy critical nonlin-earity. By the scaling and compactness arguments, we also discuss asymptotic profiles and concentration properties of the blow-up solutions.In Chapter4, we study the limit behavior of the Cauchy problem for the non-linear Schrodinger equation with time-oscillating nonlinearity and dissipation:iut+â–³u+φ(ωt)|u|αu+iζ(ωt)u=0. Under some conditions we show that as ωâ†'∞, the solution uω will locally converge to the solution of the averaged equation iut+â–³u+φ0|u|αu+iζ0u=0with the same initial condition in Lq((0.l), Br,2s) for all admissible pairs (q, r), where l∈(0, T*). We also show that if the dissipation coefficient ζ0large enough, then uω is global if ω is sufficiently large and uω converges to u in Lq ((0,∞), Br,2s), for all admissible pairs (q, r). In particular, our results hold for both subcritical and critical nonlinearities.In Chapter5, we consider the limit behavior as εâ†'0for the solution of the Cauchy problem of the nonlinear Schrodinger equation including nonlinear loss/gain with variable coefficient:iut+â–³u+λ|u|αu+iεa(t)|u|pu=0. Under some conditions we show that the solution will locally converge to the solution of the limiting equation iut+â–³u+λ|u|αu=0with the same initial data in Lq((0, l), Wl,r) for all admissible pairs (q, r), where l∈(0, T*). We also show that if the limiting solution u is global and has some decay property as tâ†'oo, then uε is global if ε is sufficiently small and uε converges to u in Lq((0,∞), Wl,r), for all admissible pairs (q, r). In particular, our results hold for both subcritical and critical nonlinearities.In Chapter6, we consider an optimal bilinear control problem for the nonlinear Schrodinger equations with singular potentials. We show well-posedness of the prob-lem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our obtained results generalize considerably the recent results of Sparber.