We consider a nonlinear Schro dinger equation in a domain 0/R n with the inhomegeneous Dirichlet boundary condition u=Q where Q is a given smooth function. The nonlinear term contributes a positive term to the energy. We prove the existence of global solutions of finite energy.
Academic Press
We present analytical methods to investigate the Cauchy problem for the complex GinzburgLandau equation u, = (v + ia)Au -(K + ip)\u\^u + yu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, K > 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when ? = 2 if (1)|0| < f * o r ( 2 ) a / ? > 0 .
For the cubic Schrodinger equation[0, oo), and Robin boundary data u,(0, t) + au(0, t) = R(t) e C 2 [0, oo) (where a is real), we show that the solution u depends continuously on « 0 andfl.
In this article we study the following nonlinear Schrödinger equation iut=Δu−g∣u∣p−1u in a domain Ω⊂Rn with initial condition u(x,0)=ϕ(x) and the Dirichlet boundary condition u(x,t)=Q(x,t) on ∂Ω, where ϕ, Q are given smooth functions. The nonlinear term contributes a negative term to the energy (i.e., g<0). We present the existence theorem for a global solution of finite energy when p⩽1+2∕n.
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