Abstract
We prove smoothing effect for solutions of a regularized Schrödinger equation on compact manifolds under the hypothesis of the Geometric control.
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L. Aloui and M. Khenissi, Stabilisation de l’équation des ondes dans un domaine extérieur,Rev. Math. Iberoamericana 18 (2002), 1–16.
C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,SIAM J. Control Optim. 305 (1992), 1024–1065.
M. Ben-Artzi and A. Devinatz, Regularity and decay of solutions to the Stark evolution equation,J. Funct. Anal. 154 (1998), 501–512.
M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation,J. Anal. Math. 58 (1992), 25–37.
N. Burq, P. Gérard, and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domain,Ann. Inst. H. Poincaré Anal. Non. Linéaire 21 (2004), 295–318.
N. Burq, Smoothing effect for Schrödinger boundaru value problems,Duke Math. J. 123 (2004), 403–427.
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,J. Amer. Math. Soc. 1 (1988), 413–439.
P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations,Indiana Univ. Math. J. 38 (1989),791–8100.
B. Dehman, P. Gérard, and G. Lebeau, Stabilization and control for the non-linear Schrödinger equation on a compact surface,Math. Z. 254 (2006), 729–749.
S. Doï, Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow,Math. Ann. 318 (2000), 355–389.
S. Doï, Remarks on the Cauchy problem for Schrödinger-type equations,Comm. Partial Differential Equations 21 (1996), 163–178.
S. Doï, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds,Duke Math. J. 82 (1996), 679–706.
R.B. Melrose, Singularities and energy decay in acoustical scattering,Duke Math. J. 46 (1979), 43–59.
J. Ralston, Solutions of the wave equation with localized energy,Comm. Pure Appl. Math. 22 (1969), 807–823.
J. Rauch, Local decay of scattering solutions of Schrödinger-type equation,J. Funct. Anal. 49 (1982), 10–56.
J. Rauch and M. Taylor, Exponential decay of solutions of hyperbolic equations in bounded domains,Indiana Univ. Math. J. 24 (1974), 79–86.
M. Reed and B. Simon,Methods of Modern Mathematical Physics I, Functional Analysis, Academic Press, New York-London, 1972.
P. Sjölin, Regularity of solutions to Schrödinger equations,Duke Math. J. 55 (1987), 699–715.
L. Vega, Schrödinger equations: pointwise convergence to the initial data,Proc. Amer. Math. Soc. 102 (1988), 874–878.
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The author is supported by Tunisian Ministry for Scientific Research and Technology, within the LAB-STI 02 program.
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Aloui, L. Smoothing effect for regularized Schrödinger equation on compact manifolds. Collect. Math. 59, 53–62 (2008). https://doi.org/10.1007/BF03191181
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DOI: https://doi.org/10.1007/BF03191181