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Well-posedness and ill-posedness for the cubic fractional Schrödinger equations
Exponential attractors for abstract equations with memory and applications to viscoelasticity
1. | Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano, Italy |
2. | Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara |
3. | Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133 |
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[2] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002. |
[4] |
I. Chueshov and I. Lasiecka, Long-Time Behavior of Second-Order Evolution Equations with Nonlinear Damping, Amer. Math. Soc., Providence, 2008.
doi: 10.1090/memo/0912. |
[5] |
M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.
doi: 10.3934/dcds.2010.27.1535. |
[6] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.
doi: 10.3934/cpaa.2005.4.705. |
[7] |
M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory, Discrete Contin. Dyn. Syst., Suppl., (2005), 200-208. |
[8] |
M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.
doi: 10.1512/iumj.2006.55.2661. |
[9] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[10] |
G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[11] |
L. Deseri, M. Fabrizio and M.J. Golden, The concept of minimal state in viscoelasticity: New free energies an applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[12] |
F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian J. Math. Phys., 15 (2008), 301-315.
doi: 10.1134/S1061920808030014. |
[13] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. |
[14] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[15] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[16] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[17] |
M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
[18] |
D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles), (French) [Reducibility of hereditary systems to dynamical systems], C.R. Acad. Sci. Paris Sér. B, 277 (1973), B471-B473. |
[19] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883.
doi: 10.1088/0951-7715/18/4/023. |
[20] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[21] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.
doi: 10.1017/S0308210509000365. |
[22] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988. |
[23] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, (French) [Dissipative dynamical systems and applications], Masson, Paris, 1991. |
[24] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV (eds. C.M. Dafermos and M. Pokorny), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[25] |
V. Pata, Exponential stability in linear viscoelasticity, Quarterly of Applied Mathematics, 64 (2006), 499-513.
doi: 10.1007/s00032-009-0098-3. |
[26] |
V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730.
doi: 10.3934/cpaa.2010.9.721. |
[27] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[28] |
M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1987. |
[29] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[30] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4684-0313-8. |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[2] |
V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273. |
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, 2002. |
[4] |
I. Chueshov and I. Lasiecka, Long-Time Behavior of Second-Order Evolution Equations with Nonlinear Damping, Amer. Math. Soc., Providence, 2008.
doi: 10.1090/memo/0912. |
[5] |
M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.
doi: 10.3934/dcds.2010.27.1535. |
[6] |
M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.
doi: 10.3934/cpaa.2005.4.705. |
[7] |
M. Conti, V. Pata and M. Squassina, Singular limit of dissipative hyperbolic equations with memory, Discrete Contin. Dyn. Syst., Suppl., (2005), 200-208. |
[8] |
M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.
doi: 10.1512/iumj.2006.55.2661. |
[9] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[10] |
G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[11] |
L. Deseri, M. Fabrizio and M.J. Golden, The concept of minimal state in viscoelasticity: New free energies an applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[12] |
F. Di Plinio and V. Pata, Robust exponential attractors for the strongly damped wave equation with memory. I, Russian J. Math. Phys., 15 (2008), 301-315.
doi: 10.1134/S1061920808030014. |
[13] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. |
[14] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbbR^3$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[15] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.
doi: 10.1017/S030821050000408X. |
[16] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[17] |
M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
[18] |
D. Fargue, Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles), (French) [Reducibility of hereditary systems to dynamical systems], C.R. Acad. Sci. Paris Sér. B, 277 (1973), B471-B473. |
[19] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata, Memory relaxation of first order evolution equations, Nonlinearity, 18 (2005), 1859-1883.
doi: 10.1088/0951-7715/18/4/023. |
[20] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138.
doi: 10.1216/RMJ-2008-38-4-1117. |
[21] |
S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.
doi: 10.1017/S0308210509000365. |
[22] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988. |
[23] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, (French) [Dissipative dynamical systems and applications], Masson, Paris, 1991. |
[24] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV (eds. C.M. Dafermos and M. Pokorny), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[25] |
V. Pata, Exponential stability in linear viscoelasticity, Quarterly of Applied Mathematics, 64 (2006), 499-513.
doi: 10.1007/s00032-009-0098-3. |
[26] |
V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730.
doi: 10.3934/cpaa.2010.9.721. |
[27] |
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. |
[28] |
M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1987. |
[29] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[30] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4684-0313-8. |
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