## Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization.(English)Zbl 1190.35206

Summary: Results of exponential/polynomial decay rates of the energy in $$L^{2}$$-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, are established. We use Kato’s theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik’s approach.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 93D15 Stabilization of systems by feedback 93B05 Controllability

### Keywords:

Schrödinger equation; exponential decay; polynomial decay
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### References:

 [1] Bisognin, E.; Bisognin, V.; Vera, O., Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping, Electron. J. differential equations, 06, 1-18, (2007) · Zbl 1387.35542 [2] Bisognin, E.; Bisognin, V.; Perla Menzala, G., Exponential stabilization of a coupled system of korteweg – de Vries equations with localized damping, Adv. differential equations, 8, 443-469, (2003) · Zbl 1057.35048 [3] Cavalcanti, M.M.; Domingos Cavalcanti, V.N.; Fukuoka, R.; Natali, F., Exponential stability for the 2-D defocusing Schrödinger equation with locally distributed damping, Differential integral equations, 22, 617-636, (2009) · Zbl 1240.35509 [4] M.M. Cavalcanti, V.N. Domingos Cavalcanti, F. Natali, Exponential decay rates for the damped Korteweg – de Vries type equation, preprint, 2008 · Zbl 1252.35242 [5] Cazenave, T., Semilinear Schrödinger equations, Courant lect. notes math., vol. 10, (2003) · Zbl 1055.35003 [6] Cazenave, T.; Weissler, F.B., The Cauchy problem for the critical nonlinear Schrödinger equation, Nonlinear anal., 14, 807-836, (1990) · Zbl 0706.35127 [7] Constantin, A.; Saut, J.-C., Local smoothing properties of dispersive equations, J. amer. math. soc., 1, 413-439, (1988) · Zbl 0667.35061 [8] Linares, F.; Ponce, G., Introduction to nonlinear dispersive equations, (2008), Springer New York [9] Linares, F.; Pazoto, A.F., On the exponential decay of the critical generalized korteweg – de Vries equation with localized damping, Proc. amer. math. soc., 135, 1515-1522, (2007) · Zbl 1107.93030 [10] Ginibre, J.; Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. math. pures appl., 64, 363-401, (1985) · Zbl 0535.35069 [11] Kato, T., On nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 46, 113-129, (1987) · Zbl 0632.35038 [12] Komornik, V., Exact controllability and stabilization. the multiplier method, (1994), Masson/John Wiley Paris · Zbl 0937.93003 [13] Massarolo, C.P.; Perla Menzala, G.; Pazoto, A.F., On the uniform decay for the korteweg – de Vries equation with weak damping, Math. methods appl. sci., 30, 1419-1435, (2007) · Zbl 1114.93080 [14] Ohta, M.; Todorova, G., Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete contin. dyn. syst. ser. A, 23, 1313-1325, (2009) · Zbl 1155.35456 [15] Pazoto, A.F., Unique continuation and decay for the korteweg – de Vries equation with localized damping, ESAIM control optim. calc. var., 11, 473-486, (2005) · Zbl 1148.35348 [16] Menzala, G.P.; Vasconcellos, C.F.; Zuazua, E., Stabilization of the korteweg – de Vries equation with localized damping, Quart. appl. math., 60, 111-129, (2002), (in English) · Zbl 1039.35107 [17] Rosier, L.; Zhang, B.-Y., Exact boundary controllability of the nonlinear Schrödinger equation, J. differential equations, 246, 4129-4153, (2009) · Zbl 1171.35015 [18] Shimomura, A., Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. partial differential equations, 31, 1407-1423, (2006) · Zbl 1105.35118 [19] Zhang, B.-Y., Unique continuation properties for the nonlinear Schrödinger equation, Proc. roy. soc. Edinburgh sect. A, 127, 191-206, (1997) [20] Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping, Comm. partial differential equations, 15, 205-235, (1990) · Zbl 0716.35010
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