# American Institute of Mathematical Sciences

• Previous Article
Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay
• DCDS Home
• This Issue
• Next Article
Regularity of center manifolds under nonuniform hyperbolicity
April  2011, 30(1): 77-113. doi: 10.3934/dcds.2011.30.77

## The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms

 1 University of Wisconsin-Madison, Mathematics Department, 480 Lincoln Dr. Madison, WI 53706-1388, United States

Received  December 2009 Revised  May 2010 Published  February 2011

In this paper, we study the generic behavior of the solutions to a large class of evolution equations. The Schrödinger evolution is considered as an application.
Citation: Sergey A. Denisov. The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 77-113. doi: 10.3934/dcds.2011.30.77
##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", National Bureau of Standards Applied Mathematics Series, 55 (). [2] A. Böttcher and B. Silbermann, "Introduction to Large Truncated Toeplitz Matrices," Springer. 1998. [3] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265. [4] M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12 (2002), 1174-1234. doi: 10.1007/s00039-002-1174-9. [5] S. Denisov, Continuous analogs of polynomials orthogonal on the unit circle. Krein systems, Int. Math. Res. Surveys, Vol. 2006 (2006). [6] S. Denisov, An evolution equation as the WKB correction in long-time asymptotics of Schrödinger dynamics, Comm. Partial Differential Equations, 33 (2008), 307-319. doi: 10.1080/03605300701249655. [7] O. Jørsboe and L. Mejlbro, "The Carleson-Hunt Theorem on Fourier Series," Lecture Notes in Mathematics 911, Springer, 1982. [8] A. Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators, Duke Math. J., 94 (1998), 619-646. doi: 10.1215/S0012-7094-98-09425-X. [9] G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411-424. [10] B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244. [11] B. Simon, "Orthogonal Polynomials on the Unit Circle," Parts 1 and 2. American Mathematical Society Colloquium Publications, 54, American Mathematical Society, Providence, RI, 2005. [12] W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations, J. Funct. Anal., 254 (2008), 2926-2946. doi: 10.1016/j.jfa.2007.11.012.

show all references

##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", National Bureau of Standards Applied Mathematics Series, 55 (). [2] A. Böttcher and B. Silbermann, "Introduction to Large Truncated Toeplitz Matrices," Springer. 1998. [3] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265. [4] M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12 (2002), 1174-1234. doi: 10.1007/s00039-002-1174-9. [5] S. Denisov, Continuous analogs of polynomials orthogonal on the unit circle. Krein systems, Int. Math. Res. Surveys, Vol. 2006 (2006). [6] S. Denisov, An evolution equation as the WKB correction in long-time asymptotics of Schrödinger dynamics, Comm. Partial Differential Equations, 33 (2008), 307-319. doi: 10.1080/03605300701249655. [7] O. Jørsboe and L. Mejlbro, "The Carleson-Hunt Theorem on Fourier Series," Lecture Notes in Mathematics 911, Springer, 1982. [8] A. Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators, Duke Math. J., 94 (1998), 619-646. doi: 10.1215/S0012-7094-98-09425-X. [9] G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411-424. [10] B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244. [11] B. Simon, "Orthogonal Polynomials on the Unit Circle," Parts 1 and 2. American Mathematical Society Colloquium Publications, 54, American Mathematical Society, Providence, RI, 2005. [12] W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations, J. Funct. Anal., 254 (2008), 2926-2946. doi: 10.1016/j.jfa.2007.11.012.
 [1] Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465 [2] Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675 [3] Mingming Chen, Xianguo Geng, Kedong Wang. Long-time asymptotics for the modified complex short pulse equation. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022060 [4] C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139 [5] Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018 [6] A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373 [7] H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119 [8] Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098 [9] Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041 [10] Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873 [11] A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185 [12] Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733 [13] Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic and Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251 [14] Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131 [15] Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005 [16] Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic and Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005 [17] Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557 [18] Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15 [19] Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017 [20] Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223

2020 Impact Factor: 1.392