Díaz Díaz, Jesús Ildefonso and Begout , Pascal (2006) On a nonlinear Schrodinger equation with a localizing effect. Comptes Rendus Mathematique, 342 (7). pp. 459-463. ISSN 1631-073X
Restricted to Repository staff only until 31 December 2020.
We consider the nonlinear Schrodinger equation associated to a singular potential of the form a vertical bar u vertical bar(-(1-m))u + bu, for some In is an element of (0, 1), on a possible unbounded domain. We use some suitable energy methods to prove that if Re(a) + Im(a) > 0 and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any t > 0. This property contrasts with the behavior of solutions associated to regular potentials (m >= 1). Related results are proved also for the associated stationary problem and for self-similar Solution on the whole space and potential a vertical bar u vertical bar(-(1-m)u). The existence of solutions is obtained by some compactness methods under additional conditions.
|Uncontrolled Keywords:||singular complex potentials|
|Subjects:||Sciences > Mathematics > Mathematical analysis|
Sciences > Mathematics > Differential geometry
S.N. Antontsev, J.I. Díaz, H.B. de Oliveira, Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem. J. Math. Fluid Mech. 6 (2004) 439–461.
S.N. Antontsev, J.I. Díaz, S. Shmarev, Energy Methods for Free Boundary Problems, Birkhäuser Boston Inc., Boston, MA, 2002.
P. Bégout, J.I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations, in press.
P. Bégout, J.I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, in press.
H. Brezis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979) 137–151.
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math., vol. 10, New York University, Courant Institute of Mathematical Science, New York, 2003.
O. Kavian, F.B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J. 41 (1993) 151–173.
B.J. LeMesurier, Dissipation at singularities of the nonlinear Schrödinger equation through limits of regularizations, Physica D 138 (2000) 334–343.
V. Liskevitch, P. Stollmann, Schrödinger operators with singular complex potentials as generators: existence and stability, Semigroup Forum 60 (2000) 337–343.
P. Rosenau, S. Schochet, Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit. Chaos 15 (2005) 1–18.
C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999.
I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monogr. Surveys Pure Appl. Math., vol. 75, Longman Scientific & Technical, Harlow, 1987.
|Deposited On:||29 May 2012 11:27|
|Last Modified:||29 May 2012 11:27|
Repository Staff Only: item control page