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Energy scattering for the focusing fractional generalized Hartree equation
Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261, Saudi Arabia |
$\left\{ \begin{array}{l}\tau {\phi _t} + N(\delta {\phi _t} + N\phi + g(\phi ) - u) = 0,\\\epsilon{u_t} + {\phi _t} + Nu = 0,\end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right)$ |
$ \tau>0 $ |
$ \delta>0 $ |
$ \epsilon\in (0,1] $ |
$ \phi $ |
$ u $ |
$ N = -\Delta:{\mathscr D}(N)\to \dot L^2(\Omega) $ |
$ \Omega\subset{\mathbb R}^d $ |
$ \Omega = \Pi_{i = 1}^d(0,L_i), $ |
$ L_i>0 $ |
$ d = 1,2 $ |
$ G(\phi) = \int_0^\phi g(\sigma)d\sigma $ |
$ j = 1 $ |
$ d = 1 $ |
$ j = 2 $ |
$ d = 2 $ |
$ g\in{\mathcal C}^{j+1}(\mathbb R) $ |
$ g'(\phi)\geq -{\mathscr C}_1 $ |
$ G(\phi)\ge -{\mathscr C}_2 $ |
$ (\phi-m(\phi))g(\phi)-{\mathscr C}_3(m(\phi))G(s)\ge -{\mathscr C}_4(m(\phi)) $ |
$ {\mathscr C}_5(\varrho)\le {\mathscr C}_l(m(\phi))\le {\mathscr C}_6(\varrho) $ |
$ l = 3,4 $ |
$ |m(\phi)|\le \varrho $ |
$ \varrho,{\mathscr C}_1, {\mathscr C}_2,{\mathscr C}_4\ge 0 $ |
$ {\mathscr C}_3, {\mathscr C}_5,{\mathscr C}_6>0 $ |
$ m(\phi) = \frac{1}{|\Omega|}\int_\Omega\phi(x)dx $ |
$ g(\phi) = \sum_{k = 1}^{2p-1}a_k\phi^k, $ |
$ p\in{\mathbb N}, $ |
$ p\ge 2, $ |
$ a_{2p-1}>0, $ |
$ {\mathcal V}_j = {\mathscr D}(N^{j/2})\times{\mathscr D}(N^{j/2}) $ |
$ \|(\psi,\varphi)\|_{{\mathcal V}_{j}} = (\|N^{j/2}\psi\|^2+m(\psi)^2+\|N^{j/2}\varphi\|^2+m(\varphi)^2)^{1/2} $ |
$ \epsilon = 0 $ |
$ \|.\|_{{\mathcal V}_{j+1}} $ |
$ \epsilon = 0 $ |
$ \|.\|_{{\mathcal V}_{j}} $ |
$ |g''(\phi)|\leq {\mathscr C}_7\left(|\phi|^{p}+1\right), $ |
$ {\mathscr C}_7>0 $ |
$ p>0 $ |
$ d = 1, 2 $ |
$ p\in [0,3] $ |
$ d = 3 $ |
$ g $ |
References:
[1] |
A. Babin and B. Nicolaenko,
Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Differ. Equ., 7 (1995), 567-589.
doi: 10.1007/BF02218725. |
[2] |
A. Bonfoh,
Dynamics of the conserved phase-field system, Appl. Anal., 95 (2016), 44-62.
doi: 10.1080/00036811.2014.997225. |
[3] |
A. Bonfoh and C. D. Enyi,
Large time behavior of a conserved phase-field system, Comm. Pure Appl. Anal., 15 (2016), 1077-1105.
doi: 10.3934/cpaa.2016.15.1077. |
[4] |
A. Bonfoh and C. D. Enyi,
The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptotic Anal., 101 (2017), 97-148.
doi: 10.3233/ASY-161395. |
[5] |
D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., vol. 296, Longman Sci. Tech., Harlow, 1993, 77–85. |
[6] |
D. Brochet, D. Hilhorst and A. Novick-Cohen,
Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns, 1 (1996), 547-568.
|
[7] |
G. Caginalp,
Conserved-phase field system: implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.
|
[8] |
L. Dung and B. Nicolaenko,
Exponential attractors in Banach spaces, J. Dyn. Differ. Equ., 13 (2001), 791-806.
doi: 10.1023/A:1016676027666. |
[9] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. |
[10] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb R^3$, C. R. Math. Acad. Sci. Paris, 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[11] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[12] |
C. M. Elliott and A. M. Stuart,
The viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differ. Equ., 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[13] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata,
A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127.
doi: 10.1090/S0002-9939-05-08340-1. |
[14] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata,
Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci., 15 (2005), 165-198.
doi: 10.1142/S0218202505000327. |
[15] |
G. Gilardi,
On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161.
|
[16] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
doi: 10.1090/surv/025. |
[17] |
J. K. Hale and G. Raugel,
Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equ., 73 (1988), 197-214.
doi: 10.1016/0022-0396(88)90104-0. |
[18] |
J. K. Hale and G. Raugel,
Lower-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dyn. Differ. Equ., 2 (1990), 19-67.
doi: 10.1007/BF01047769. |
[19] |
A. Miranville,
Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 145-150.
doi: 10.1016/S0764-4442(99)80153-0. |
[20] |
A. Miranville,
On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.
doi: 10.1016/j.jmaa.2012.11.038. |
[21] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handb. Differ. Equ., 4 (2018), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[22] |
G. Mola,
Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., 7 (2008), 317-353.
doi: 10.3934/cpaa.2008.7.317. |
[23] |
G. Mola,
Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Models Methods Appl. Sci., 32 (2009), 2368-2404.
doi: 10.1002/mma.1139. |
[24] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Univ. Press, New York, 1988. |
[25] |
G. Raugel, Singularly perturbed hyperbolic equations revisited, in International Conference on Differential Equations, World Sci. Publishing, River Edge, NJ, 2000. |
[26] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
show all references
References:
[1] |
A. Babin and B. Nicolaenko,
Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Differ. Equ., 7 (1995), 567-589.
doi: 10.1007/BF02218725. |
[2] |
A. Bonfoh,
Dynamics of the conserved phase-field system, Appl. Anal., 95 (2016), 44-62.
doi: 10.1080/00036811.2014.997225. |
[3] |
A. Bonfoh and C. D. Enyi,
Large time behavior of a conserved phase-field system, Comm. Pure Appl. Anal., 15 (2016), 1077-1105.
doi: 10.3934/cpaa.2016.15.1077. |
[4] |
A. Bonfoh and C. D. Enyi,
The Cahn-Hilliard equation as limit of a conserved phase-field system, Asymptotic Anal., 101 (2017), 97-148.
doi: 10.3233/ASY-161395. |
[5] |
D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., vol. 296, Longman Sci. Tech., Harlow, 1993, 77–85. |
[6] |
D. Brochet, D. Hilhorst and A. Novick-Cohen,
Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns, 1 (1996), 547-568.
|
[7] |
G. Caginalp,
Conserved-phase field system: implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.
|
[8] |
L. Dung and B. Nicolaenko,
Exponential attractors in Banach spaces, J. Dyn. Differ. Equ., 13 (2001), 791-806.
doi: 10.1023/A:1016676027666. |
[9] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Masson, Paris, 1994. |
[10] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb R^3$, C. R. Math. Acad. Sci. Paris, 330 (2000), 713-718.
doi: 10.1016/S0764-4442(00)00259-7. |
[11] |
M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[12] |
C. M. Elliott and A. M. Stuart,
The viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differ. Equ., 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[13] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata,
A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127.
doi: 10.1090/S0002-9939-05-08340-1. |
[14] |
S. Gatti, M. Grasselli, A. Miranville and V. Pata,
Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math. Models Methods Appl. Sci., 15 (2005), 165-198.
doi: 10.1142/S0218202505000327. |
[15] |
G. Gilardi,
On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161.
|
[16] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
doi: 10.1090/surv/025. |
[17] |
J. K. Hale and G. Raugel,
Upper-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differ. Equ., 73 (1988), 197-214.
doi: 10.1016/0022-0396(88)90104-0. |
[18] |
J. K. Hale and G. Raugel,
Lower-semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dyn. Differ. Equ., 2 (1990), 19-67.
doi: 10.1007/BF01047769. |
[19] |
A. Miranville,
Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 145-150.
doi: 10.1016/S0764-4442(99)80153-0. |
[20] |
A. Miranville,
On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.
doi: 10.1016/j.jmaa.2012.11.038. |
[21] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handb. Differ. Equ., 4 (2018), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[22] |
G. Mola,
Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., 7 (2008), 317-353.
doi: 10.3934/cpaa.2008.7.317. |
[23] |
G. Mola,
Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Models Methods Appl. Sci., 32 (2009), 2368-2404.
doi: 10.1002/mma.1139. |
[24] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Univ. Press, New York, 1988. |
[25] |
G. Raugel, Singularly perturbed hyperbolic equations revisited, in International Conference on Differential Equations, World Sci. Publishing, River Edge, NJ, 2000. |
[26] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
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