Abstract
We consider a nonlinear Schrödinger equation in a time-dependent domain Q τ of ℝ2 given by
We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.
Similar content being viewed by others
References
Antunes, G.O., Silva, M.D.G., Apolaya, R.F.: Schrödinger equations in non cylindrical domains-exact controllability. Differ. Integral Equ. 11(5), 755–770 (1998)
Barab, J.E.: Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys. 25(11), 3270–3273 (1984)
Bernardi, M.L., Bonfanti, G., Lutteroti, F.: Abstract Schrödinger type differential equations with variable domain. J. Math. Anal. Appl. 211, 84–105 (1997)
Bisognin, E., Bisognin, V., Sepúlveda, M., Vera, O.: Coupled system of Korteweg de Vries equations type in domains with moving boundaries. J. Comput. Appl. Math. 220, 290–321 (2008)
Cavalcanti, M.M., Cavalcanti, V.N.D., Natali, F.M., Soriano, J.A.: Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization. J. Differ. Equ. 248, 2955–2971 (2010)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Doronin, G., Larkin, N.: KDV equation in domains with moving boundaries. J. Math. Anal. Appl. 48, 157–172 (2007)
Ferreira, J., Benabidallah, R., Muñoz Rivera, J.E.: Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain. Rend. Mat. Ser. VII 19(1), 177–193 (1999)
Ghidaglia, J.M.: Finite dimensional behavior damped driven Schrödinger equations. Ann. Inst. Henri Poincaré 5(4), 365–405 (1988)
Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations I. The Cauchy problem. General case. J. Funct. Anal. 32, 1–32 (1979)
Kenig, C.E., Ponce, G., Vega, L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 10(3), 255–288 (1993)
Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires. Gauthiers-Villars, Paris (1969)
Miranda, M.M., Medeiros, L.A.: Contrôlabilité exacte de l’équation de Schrödinger dans des domaines non cylindriques. C. R. Acad. Sci. Paris 319, 685–689 (1994)
Teman, R.: Sur um probléme non linéaire. J. Math. Pures Appl. 48, 159–172 (1969)
Acknowledgements
Mauricio Sepúlveda thanks the support of Fondecyt project 1110540, CONICYT project Anillo ACT1118 (ANANUM), and Basal, CMM, Universidad de Chile. Octavio Vera thanks the support of Fondecyt project 1121120. V. Bisognin, C. Buriol and Marcio Ferreira thank the support of Fundação de Amparo à pesquisa do Estado do Rio Grande do Sul. FAPERGS.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bisognin, V., Buriol, C., Ferreira, M.V. et al. Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries. Acta Appl Math 125, 159–172 (2013). https://doi.org/10.1007/s10440-012-9785-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-012-9785-0