Asymptotic Behaviour for a Nonlinear Schr枚dinger Equation in Domains with Moving Boundaries

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• 作者：Vanilde Bisognin (1)
  Celene Buriol (2)
  Marcio V. Ferreira (2)
  Mauricio Sep煤lveda (3)
  Octavio Vera (4)
  
• 关键词：Schr枚dinger equation ; Stabilization ; Moving boundary ; 35K60 ; 93C20
• 刊名：Acta Applicandae Mathematicae
• 出版年：2013
• 出版时间：June 2013
• 年：2013
• 卷：125
• 期：1
• 页码：159-172
• 全文大小：521KB
• 参考文献：1. Antunes, G.O., Silva, M.D.G., Apolaya, R.F.: Schr枚dinger equations in non cylindrical domains-exact controllability. Differ. Integral Equ. <strong class="a-plus-plus">11strong>(5), 755鈥?70 (1998)
  2. Barab, J.E.: Nonexistence of asymptotically free solutions for a nonlinear Schr枚dinger equation. J. Math. Phys. <strong class="a-plus-plus">25strong>(11), 3270鈥?273 (1984) ss="external" href="http://dx.doi.org/10.1063/1.526074">CrossRef
  3. Bernardi, M.L., Bonfanti, G., Lutteroti, F.: Abstract Schr枚dinger type differential equations with variable domain. J. Math. Anal. Appl. <strong class="a-plus-plus">211strong>, 84鈥?05 (1997) ss="external" href="http://dx.doi.org/10.1006/jmaa.1997.5422">CrossRef
  4. Bisognin, E., Bisognin, V., Sep煤lveda, M., Vera, O.: Coupled system of Korteweg de Vries equations type in domains with moving boundaries. J. Comput. Appl. Math. <strong class="a-plus-plus">220strong>, 290鈥?21 (2008) ss="external" href="http://dx.doi.org/10.1016/j.cam.2007.08.008">CrossRef
  5. Cavalcanti, M.M., Cavalcanti, V.N.D., Natali, F.M., Soriano, J.A.: Qualitative aspects for the cubic nonlinear Schr枚dinger equations with localized damping: exponential and polynomial stabilization. J. Differ. Equ. <strong class="a-plus-plus">248strong>, 2955鈥?971 (2010) ss="external" href="http://dx.doi.org/10.1016/j.jde.2010.03.023">CrossRef
  6. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
  7. Doronin, G., Larkin, N.: KDV equation in domains with moving boundaries. J. Math. Anal. Appl. <strong class="a-plus-plus">48strong>, 157鈥?72 (2007)
  8. Ferreira, J., Benabidallah, R., Mu帽oz Rivera, J.E.: Asymptotic behaviour for the nonlinear beam equation in a time-dependent domain. Rend. Mat. Ser. VII <strong class="a-plus-plus">19strong>(1), 177鈥?93 (1999)
  9. Ghidaglia, J.M.: Finite dimensional behavior damped driven Schr枚dinger equations. Ann. Inst. Henri Poincar茅 <strong class="a-plus-plus">5strong>(4), 365鈥?05 (1988)
  10. Ginibre, J., Velo, G.: On a class of nonlinear Schr枚dinger equations聽I. The Cauchy problem. General case. J. Funct. Anal. <strong class="a-plus-plus">32strong>, 1鈥?2 (1979) ss="external" href="http://dx.doi.org/10.1016/0022-1236(79)90076-4">CrossRef
  11. Kenig, C.E., Ponce, G., Vega, L.: Small solutions to nonlinear Schr枚dinger equations. Ann. Inst. Henri Poincar茅 <strong class="a-plus-plus">10strong>(3), 255鈥?88 (1993)
  12. Lions, J.L.: Quelques M茅thodes de R茅solution des Probl茅mes aux Limites Non Lin茅aires. Gauthiers-Villars, Paris (1969)
  13. Miranda, M.M., Medeiros, L.A.: Contr么labilit茅 exacte de l鈥櫭﹒uation de Schr枚dinger dans des domaines non cylindriques. C. R. Acad. Sci. Paris <strong class="a-plus-plus">319strong>, 685鈥?89 (1994)
  14. Teman, R.: Sur um probl茅me non lin茅aire. J. Math. Pures Appl. <strong class="a-plus-plus">48strong>, 159鈥?72 (1969)
  
• 作者单位：Vanilde Bisognin (1)
  Celene Buriol (2)
  Marcio V. Ferreira (2)
  Mauricio Sep煤lveda (3)
  Octavio Vera (4)
  
  1. Centro Universitario Franciscano, 97010-032, Santa Maria, RS, Brazil
  2. Departamento de Matem谩tica, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
  3. CI虏MA and Departamento de Ingenier铆a Matem谩tica, Universidad de Concepci贸n, Concepci贸n, Chile
  4. Departamento de Matem谩tica, Universidad del B铆o-B铆o, Collao 1202, Casilla 5-C, Concepci贸n, Chile
  
• ISSN：1572-9036

文摘

We consider a nonlinear Schr枚dinger equation in a time-dependent domain Q 蟿 sub> of 鈩?sup class="a-plus-plus">2 given by $$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$ We prove the well-posedness of the above model and analyze the behaviour of the solution as t鈫?鈭? We consider two situations: the conservative case (纬=0) and the dissipative case (纬>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.