Uniform decay rates for the energy of weakly damped defocusing semilinear Schrodinger equations with inhomogeneous Dirichlet boundary control

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:251

Article

Ozsari, Turker¹  Kalantarov, Varga K.²  Lasiecka, Irena^3,4 

[1] Dogus Univ, Dept Math, TR-34722 Istanbul, Turkey

[2] Koc Univ, Dept Math, TR-34450 Istanbul, Turkey

[3] Univ Virginia, Dept Math, Charlottesville, VA 22904 USA

[4] Polish Acad Sci, Syst Res Inst, PL-01447 Warsaw, Poland

关键词: Nonlinear Schrodinger equation;    Inhomogeneous Dirichlet boundary condition;    Existence;    Stabilization;    Boundary control;    Direct multiplier method;    Monotone operator theory;    Compactness;   

DOI  :  10.1016/j.jde.2011.04.003

来源: Elsevier

【 摘 要 】

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped de-focusing semilinear Schrodinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H-1-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H-2-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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