Well-posedness for degenerate Schrödinger equations

Massimo Cicognani; Michael Reissig

doi:10.3934/eect.2014.3.15

Article Contents

2014, Volume 3, Issue 1: 15-33. Doi: 10.3934/eect.2014.3.15

This issue Previous Article Existence and asymptotic behaviour for solutions of dynamical equilibrium systems Next Article Optimal control for stochastic heat equation with memory

Well-posedness for degenerate Schrödinger equations

Massimo Cicognani^1, and
Michael Reissig^2,

1.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna
2.
TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte Analysis, 09596 Freiberg

Received: February 28, 2013

Revised: July 31, 2013

Published: February 2014

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Abstract

We consider the initial value problem for Schrödinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.

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Mathematics Subject Classification: Primary: 35J10; Secondary: 35Q41.

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References

[1]	A. Ascanelli, M. Cicognani and F. Colombini, The global Cauchy problem for a vibrating beam equation, Journal of Differential Equations, 247 (2009), 1440-1451.doi: 10.1016/j.jde.2009.06.012.
[2]	A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation, Rend. Semin. Mat. Torino, 67 (2009), 151-164.
[3]	M. Cicognani and F. Colombini, Optimal well-posedness of the cauchy problem for evolution equations with $C^N$ coefficients, Differential and Integral Equations, 17 (2004), 1079-1092.
[4]	M. Cicognani and T. Herrmanni, $H^\infty$ well-posedness for a $2$-evolution Cauchy problem with complex coefficients, Journal of Pseudo-Differential Operators and Applications, 4 (2013), 63-90.doi: 10.1007/s11868-013-0062-4.
[5]	S. I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.
[6]	S. I. Doi, Remarks on the Cauchy problem for Schrödingerr-type equations, Comm. Partial Differential Equations, 21 (1996), 163-178.doi: 10.1080/03605309608821178.
[7]	M. Dreher, Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces, Bull. Sci. Math., 127 (2003), 485-503.doi: 10.1016/S0007-4497(03)00026-5.
[8]	W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581.
[9]	W. Ichinose, Sufficient condition on $H^\infty$ well-posedness for Schrödinger type equations, Comm. Partial Differential Equations, 9 (1984), 33-48.doi: 10.1080/03605308408820324.
[10]	W. Ichinose, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term, J. Math. Kyoto Univ., 33 (1993), 647-663.
[11]	W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620.
[12]	K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), 179-202.doi: 10.2969/jmsj/05010179.
[13]	K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math., 119 (1995), 459-473.
[14]	K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.
[15]	S. Mizohata, On some Schrödinger type equations, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 81-84.doi: 10.3792/pjaa.57.81.