The paper considers the initial value problem of a general type of nonlinear Schrödinger equations
$ iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x) $
posed on a finite domain $ x\in [0, L] $ with an $ L^2 $-stabilizing feedback control law $ u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t), $ where $ L>0 $, $ \alpha, \beta $ are real constants with $ \alpha\beta<0 $ and $ \beta\neq \pm 1 $, and $ f(u) $ is a smooth function from $ \mathbb{C} $ to $ \mathbb{C} $ satisfying some growth conditions. It is shown that for $ s \in \left ( \frac12, 1\right ] $ and $ w_0 (x) \in H^s(0, L ) $ with the boundary conditions described above, the problem is locally well-posed for $ u \in C([0, T]; H^s (0, L )) $. Moreover, the solution with small initial condition exists globally and approaches to 0 as $ t \rightarrow + \infty $.
Citation: |
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