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Mathematics of traveling waves in chemotaxis --Review paper--
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing
1. | Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, United States |
References:
[1] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland, Amsterdam, 1992. |
[3] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[4] |
P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[6] |
J. Bell, Some threshold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190.
doi: 10.1016/0025-5564(81)90085-7. |
[7] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. |
[8] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynamical Systems B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[9] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443
doi: 10.3934/dcds.2008.21.415. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[11] |
T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis, 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[12] |
I. Chueshow, "Monotone Random Systems - Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[13] |
I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.
doi: 10.1080/1468936042000207792. |
[14] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[15] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[16] |
J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[17] |
J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[18] |
J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. |
[19] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. |
[20] |
F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. |
[21] |
M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[22] |
M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[23] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[24] |
J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete and Continuous Dynamical Systems, 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[25] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[26] |
Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for infinite-dimensional random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp.
doi: 10.1090/S0065-9266-10-00574-0. |
[27] |
Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems, Nonlinear Analysis, 54 (2003), 873-884.
doi: 10.1016/S0362-546X(03)00112-3. |
[28] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.
doi: 10.1137/0520057. |
[29] |
M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326.
doi: 10.1016/0022-247X(89)90043-7. |
[30] |
S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp. |
[31] |
J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070. |
[32] |
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192. |
[33] |
R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[34] |
Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions, J. Differential Equations, 144 (1998), 1-43.
doi: 10.1006/jdeq.1997.3383. |
[35] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[36] |
B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[37] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[38] |
B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[39] |
B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, 71 (2009), 2811-2828.
doi: 10.1016/j.na.2009.01.131. |
[40] |
B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[41] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp. |
[42] |
B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electronic Journal of Differential Equations, 2012 (2012), 18 pp. |
[43] |
B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations,, submitted for publication., ().
|
show all references
References:
[1] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Translated and revised from the 1989 Russian original by Babin, Studies in Mathematics and its Applications, 25, North-Holland, Amsterdam, 1992. |
[3] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonl. Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[4] |
P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[6] |
J. Bell, Some threshold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190.
doi: 10.1016/0025-5564(81)90085-7. |
[7] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. |
[8] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynamical Systems B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[9] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443
doi: 10.3934/dcds.2008.21.415. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[11] |
T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis, 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[12] |
I. Chueshow, "Monotone Random Systems - Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[13] |
I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.
doi: 10.1080/1468936042000207792. |
[14] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[15] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[16] |
J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[17] |
J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[18] |
J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. |
[19] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. |
[20] |
F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. |
[21] |
M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[22] |
M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[23] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[24] |
J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete and Continuous Dynamical Systems, 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[25] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[26] |
Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for infinite-dimensional random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206 (2010), vi+106 pp.
doi: 10.1090/S0065-9266-10-00574-0. |
[27] |
Y. Lu and Z. Shao, Determining nodes for partly dissipative reaction diffusion systems, Nonlinear Analysis, 54 (2003), 873-884.
doi: 10.1016/S0362-546X(03)00112-3. |
[28] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.
doi: 10.1137/0520057. |
[29] |
M. Marion, Inertial manifolds associated to partly dissipative reaction-diffusion systems, J. Math. Anal. Appl., 143 (1989), 295-326.
doi: 10.1016/0022-247X(89)90043-7. |
[30] |
S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196 (2008), vi+105 pp. |
[31] |
J. Nagumo, S. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1964), 2061-2070. |
[32] |
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192. |
[33] |
R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[34] |
Z. Shao, Existence of inertial manifolds for partly dissipative reaction diffusion systems in higher space dimensions, J. Differential Equations, 144 (1998), 1-43.
doi: 10.1006/jdeq.1997.3383. |
[35] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[36] |
B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[37] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[38] |
B. Wang, Random attractors for the Stochastic Benjamin-Bona-Mahony Equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.
doi: 10.1016/j.jde.2008.10.012. |
[39] |
B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, 71 (2009), 2811-2828.
doi: 10.1016/j.na.2009.01.131. |
[40] |
B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[41] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 18 pp. |
[42] |
B. Wang, Periodic random attractors for stochastic Navier-Stokes equations on unbounded domains, Electronic Journal of Differential Equations, 2012 (2012), 18 pp. |
[43] |
B. Wang, Existence, stability and bifurcation of random periodic solutions of stochastic parabolic equations,, submitted for publication., ().
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