Dynamics of multi-valued retarded \(p\)-Laplace equations driven by nonlinear colored noise. (English) Zbl 1531.35171
Summary: This paper mainly considers the long-term behavior of \(p\)-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved.
©2023 American Institute of Physics
©2023 American Institute of Physics
MSC:
| 35L55 | Higher-order hyperbolic systems |
| 35L30 | Initial value problems for higher-order hyperbolic equations |
| 35B41 | Attractors |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
References:
| [1] | Arnold, L., Random Dynamical Systems (1998), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0906.34001 |
| [2] | Aubin, J. P.; Frankowska, H., Set-Valued Analysis (1990), Birkhäuser · Zbl 0713.49021 |
| [3] | Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuss, B.; Valero, J., Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst. A, 21, 415-443 (2008) · Zbl 1155.60025 · doi:10.3934/dcds.2008.21.415 |
| [4] | Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuss, B.; Valero, J., Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. B, 14, 439-455 (2010) · Zbl 1201.60063 · doi:10.3934/dcdsb.2010.14.439 |
| [5] | Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuß, B.; Valero, J., Global attractor for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal.: Theory, Methods Appl., 73, 183-201 (2010) · Zbl 1194.35071 · doi:10.1016/j.na.2010.03.012 |
| [6] | Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuss, B.; Valero, J., Attractors for a random evolution equation with infinite memory: Theoretical results, Discrete Contin. Dyn. Syst. B, 22, 1779-1800 (2017) · Zbl 1359.60080 · doi:10.3934/dcdsb.2017106 |
| [7] | Caraballo, T.; Herrera-Cobos, M.; Marín-Rubio, P., Asymptotic behaviour of nonlocal p-Laplacian reaction-diffusion problems, J. Math. Anal. Appl., 459, 997-1015 (2018) · Zbl 1483.35115 · doi:10.1016/j.jmaa.2017.11.013 |
| [8] | Caraballo, T.; Marín-Rubio, P.; Valero, J., Attractors for differential equations with unbounded delays, J. Differ. Equations, 239, 311-342 (2007) · Zbl 1135.34040 · doi:10.1016/j.jde.2007.05.015 |
| [9] | Chen, P.; Wang, B.; Wang, R.; Zhang, X., Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains, Math. Ann., 386, 343-373 (2022) · Zbl 1520.35012 · doi:10.1007/s00208-022-02400-0 |
| [10] | Chen, P.; Wang, R.; Zhang, X., Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains, Bull. Sci. Math., 173, 103071 (2021) · Zbl 1506.35262 · doi:10.1016/j.bulsci.2021.103071 |
| [11] | Chen, P.; Zhang, X., Random dynamics of stochastic BBM equations driven by nonlinear colored noise on unbounded channel, J. Evol. Equations, 22, 87 (2022) · Zbl 1501.35081 · doi:10.1007/s00028-022-00845-z |
| [12] | Damascelli, L., Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. Henri Poincare, Sect. C, Anal. Non Linéaire, 15, 493-516 (1998) · Zbl 0911.35009 · doi:10.1016/s0294-1449(98)80032-2 |
| [13] | Esteban, J. R.; Vazquez, J. L., On the equation of turbulent filtration in one- dimensional porous media, Nonlinear Anal.: Theory, Methods Appl., 10, 1303-1325 (1986) · Zbl 0613.76102 · doi:10.1016/0362-546x(86)90068-4 |
| [14] | Geredeli, P. G.; Khanmamedov, A., Long-time dynamics of the parabolic p-Laplacian equation, Commun. Pure Appl. Anal., 12, 735-754 (2013) · Zbl 1267.35043 · doi:10.3934/cpaa.2013.12.735 |
| [15] | Gu, A.; Wang, B., Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. B, 23, 1689-1720 (2018) · Zbl 1396.35078 · doi:10.3934/dcdsb.2018072 |
| [16] | Gu, A.; Wang, B., Random attractors of reaction-diffusion equations without uniqueness driven by nonlinear colored noise, J. Math. Anal. Appl., 486, 123880 (2020) · Zbl 1440.35350 · doi:10.1016/j.jmaa.2020.123880 |
| [17] | Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21, 11-41 (1978) · Zbl 0383.34055 |
| [18] | Hino, Y.; Murakami, S.; Naito, T., Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics Vol. 1473 (1991), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0732.34051 |
| [19] | Kappel, F.; Schappacher, W., Some considerations to the fundamental theory of infinite delay equations, J. Differ. Equations, 37, 141-183 (1980) · Zbl 0466.34036 · doi:10.1016/0022-0396(80)90093-5 |
| [20] | Keller, H. B.; Cohen, D. S., Some positone problems suggested by nonlinear heat generation, Indiana Univ. Math. J., 16, 1361-1376 (1967) · Zbl 0152.10401 · doi:10.1512/iumj.1967.16.16087 |
| [21] | Kłosek-Dygas, M. M.; Matkowsky, B. J.; Schuss, Z., Colored noise in dynamical systems, SIAM J. Appl. Math., 48, 425-441 (1988) · Zbl 0638.60086 · doi:10.1137/0148023 |
| [22] | Krause, A.; Lewis, M.; Wang, B., Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246, 365-376 (2014) · Zbl 1338.35514 · doi:10.1016/j.amc.2014.08.033 |
| [23] | Krause, A.; Wang, B., Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417, 1018-1038 (2014) · Zbl 1307.35059 · doi:10.1016/j.jmaa.2014.03.037 |
| [24] | Li, Y.; Gu, A.; Li, J., Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equations, 258, 504-534 (2015) · Zbl 1306.37091 · doi:10.1016/j.jde.2014.09.021 |
| [25] | Li, Y.; Yin, J., Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations, Discrete Contin. Dyn. Syst. S, 9, 1939-1957 (2016) · Zbl 1353.37145 · doi:10.3934/dcdss.2016079 |
| [26] | Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires (1969), Dunod: Dunod, Paris · Zbl 0189.40603 |
| [27] | Liu, L.; Caraballo, T.; Marín-Rubio, P., Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differ. Equations, 265, 5685-5708 (2018) · Zbl 1403.35206 · doi:10.1016/j.jde.2018.07.008 |
| [28] | Liu, L.; Fu, X., Existence and upper semi-continuity of pullback attractors of a p-Laplacian equation with delay, J. Math. Phys., 58, 082702 (2017) · Zbl 1375.35051 · doi:10.1063/1.5000076 |
| [29] | Marín-Rubio, P.; Real, J.; Valero, J., Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal.: Theory, Methods Appl., 74, 2012-2030 (2011) · Zbl 1218.35045 · doi:10.1016/j.na.2010.11.008 |
| [30] | Samprogna, R. A.; Caraballo, T., Pullback attractor for a dynamic boundary non-autonomous problem with infinite delay, Discrete Contin. Dyn. Syst. B, 23, 509-523 (2018) · Zbl 1415.35058 · doi:10.3934/dcdsb.2017195 |
| [31] | Samprogna, R. A.; Schiabel, K.; Gentile Moussa, C. B., Pullback attractors for multivalued processes and application to nonautonomous problems with dynamic boundary conditions, Set-Valued Var. Anal., 27, 19-50 (2019) · Zbl 1473.35062 · doi:10.1007/s11228-017-0404-0 |
| [32] | Showalter, R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (1997), American Mathematical Society: American Mathematical Society, Providence · Zbl 0870.35004 |
| [33] | Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences Vol. 68 (1997), Springer-Verlag: Springer-Verlag, New York · Zbl 0871.35001 |
| [34] | Wang, B., Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. B, 22, 2011-2051 (2017) · Zbl 1359.35012 · doi:10.3934/dcdsb.2017119 |
| [35] | Wang, Y.; Kloeden, P. E., Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Anal.: Theory, Methods Appl., 90, 86-95 (2013) · Zbl 1282.35398 · doi:10.1016/j.na.2013.05.026 |
| [36] | Wang, Y.; Wang, J., Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equations, 259, 728-776 (2015) · Zbl 1320.35091 · doi:10.1016/j.jde.2015.02.026 |
| [37] | Xu, J.; Caraballo, T.; Valero, J., Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion, J. Differ. Equations, 327, 418-447 (2022) · Zbl 1489.35018 · doi:10.1016/j.jde.2022.04.033 |
| [38] | Xu, J.; Caraballo, T.; Valero, J., Asymptotic behavior of nonlocal partial differential equations with long time memory, Discrete Contin. Dyn. Syst. S, 15, 3059-3078 (2022) · Zbl 1498.35088 · doi:10.3934/dcdss.2021140 |
| [39] | Xu, J.; Zhang, Z.; Caraballo, T., Non-autonomous nonlocal partial differential equations with delay and memory, J. Differ. Equations, 270, 505-546 (2021) · Zbl 1451.35072 · doi:10.1016/j.jde.2020.07.037 |
| [40] | Zhang, P.; Gu, A., Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms, Stochastics Dyn., 22, 2140013 (2022) · Zbl 1498.37120 · doi:10.1142/s021949372140013x |
| [41] | Zhao, W.-Q., Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput., 239, 358-374 (2014) · Zbl 1334.35456 · doi:10.1016/j.amc.2014.04.106 |
| [42] | Zhao, W.-Q., High-order Wong-Zakai approximations for non-autonomous stochastic p-Laplacian equations on \(\mathbb{R}^n\), Commun. Pure Appl. Anal., 20, 243-280 (2021) · Zbl 1460.35404 · doi:10.3934/cpaa.2020265 |
| [43] | Zhu, K.; Xie, Y.; Zhou, F.; Zhou, Q., Pullback attractors for p-Laplacian equations with delays, J. Math. Phys., 62, 022702 (2021) · Zbl 1460.35182 · doi:10.1063/1.5126618 |
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