Asymptotic behavior of fractional stochastic heat equations in materials with memory

References

• Dong J, Xu M. Space-time fractional Schrödinger equation with time-independent potentials. J Math Anal Appl. 2008;344:1005–1017. doi: 10.1016/j.jmaa.2008.03.061
   Web of Science ®Google Scholar
• Guo B, Han Y, Xin J. Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl Math Comput. 2008;204:458–477.
   Web of Science ®Google Scholar
• Guo B, Huo Z. Global well-posedness for the fractional nonlinear Schrödinger equation. Commun Partial Differ Equ. 2011;36:247–255. doi: 10.1080/03605302.2010.503769
   Web of Science ®Google Scholar
• Guo B, Zeng M. Solutions for the fractional Landau–Lifshitz equation. J Math Anal Appl. 2010;361:131–138. doi: 10.1016/j.jmaa.2009.09.009
   Web of Science ®Google Scholar
• Pu X, Guo B. Global weak solutions of the fractional Landau–Lifshitz–Maxwell equation. J Math Anal Appl. 2010;372:86–98. doi: 10.1016/j.jmaa.2010.06.035
   Web of Science ®Google Scholar
• Pu X, Guo B. Well-posedness and dynamics for the fractional Ginzburg–Laudau equation. Appl Anal. 2013;92:318–334. doi: 10.1080/00036811.2011.614601
   Web of Science ®Google Scholar
• Tarasov Vasily E, Zaslavsky George M. Fractional Ginzburg–Laudau equation for fractal media. Phys A. 2005;354:249–261. doi: 10.1016/j.physa.2005.02.047
   Web of Science ®Google Scholar
• Gal C, Warma M. Reaction–diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete Contin Dyn Syst. 2016;36:1279–1319. doi: 10.3934/dcds.2016.36.1279
   Web of Science ®Google Scholar
• Wang B. Asymptotic behavior of non-autonomous fractional stochastic reaction–diffusion equations. Nonlinear Anal TMA. 2017;158:60–82. doi: 10.1016/j.na.2017.04.006
   Web of Science ®Google Scholar
• Lv G, Duan J. Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals. J Math Anal Appl. 2017;449:176–194. doi: 10.1016/j.jmaa.2016.12.011
   Web of Science ®Google Scholar
• Lv G, Gao H, Wei J. BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations. J Differ Equ. 2019;266:2666–2717. doi: 10.1016/j.jde.2018.08.042
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. On the spectrum of two different fractional operators. Proc R Soc Edinburgh Sect A. 2014;144:831–855. doi: 10.1017/S0308210512001783
   Web of Science ®Google Scholar
• Hale JK. Asymptotic behavior of dissipative systems. In: Mathematical Surveys and Monographs. vol. 25, AMS, Providence, RI; 1988.
   Google Scholar
• Robinson JC. Infinite-dimensional dynamical systems. Cambridge: Cambridge University Press; 2001.
   Google Scholar
• Sell G, You Y. Dynamics of evolutional equations. New York: Springer-Verlag; 2002.
   Google Scholar
• Temam R. Infinite dimension dynamical systems in mechanics and physics. New York: Springer-Verlag; 1995.
   Google Scholar
• Arnold L. Random dynamical systems. New York: Springer-Verlag; 1998.
   Google Scholar
• Bates PW, Lisei H, Lu K. Attractors for stochastic lattice dynamical system. Stoch Dyn. 2006;6:1–21. doi: 10.1142/S0219493706001621
   Web of Science ®Google Scholar
• Chueshov I. Monotone random systems theory and applications. Berlin, Heidelbrg: Springer-Verlag; 2002.
   Google Scholar
• Crauel H, Flandoli F. Attractors for random dynamical systems. Probab Theory Related Fields. 1994;100:365–393. doi: 10.1007/BF01193705
   Web of Science ®Google Scholar
• Crauel H, Debussche A, Flandoli F. Random attractors. J Dyn Differ Equ. 1997;9:307–341. doi: 10.1007/BF02219225
   Google Scholar
• Fan X, Wang Y. Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping. Phys Lett A. 2007;365:17–27. doi: 10.1016/j.physleta.2006.12.045
   Web of Science ®Google Scholar
• Flandoli F, Schmalfuss B. Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise. Stoch Stoch Rep. 1996;59:21–45. doi: 10.1080/17442509608834083
   Google Scholar
• Schmalfuss B. Backward cocycle and atttractors of stochastic differential equations. In: Reitmann V, Riedrich T, Koksch N, editors. International semilar on applied mathematics-nonlinear dynamics: attractor approximation and global behavior. Dresden: Technishe Universität; 1992, p. 185–192.
   Google Scholar
• Zhao C, Zhou S. Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications. J Math Anal Appl. 2009;354:78–95. doi: 10.1016/j.jmaa.2008.12.036
   Web of Science ®Google Scholar
• Bates PW, Lu K, Wang B. Random attractors for stochastic reaction–diffusion equations on unbounded domains. J Differ Equ. 2009;246:845–869. doi: 10.1016/j.jde.2008.05.017
   Web of Science ®Google Scholar
• Brzezniak Z, Li Y. Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains. Trans Am Math Soc. 2006;358:5587–5629. doi: 10.1090/S0002-9947-06-03923-7
   Web of Science ®Google Scholar
• Garrido-Atienza M, Lu K, Schmalfuss B. Random dynamical systems for stochastic equations driven by a fractional Brownian motion. Discrete Contin Dyn Syst Ser B. 2010;14:473–493.
   Web of Science ®Google Scholar
• Maslowski B, Schmalfuss B. Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch Anal Appl. 2004;22:1577–1607. doi: 10.1081/SAP-200029498
   Web of Science ®Google Scholar
• Shen Z, Zhou S, Shen W. One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J Differ Equ. 2010;248:1432–1457. doi: 10.1016/j.jde.2009.10.007
   Web of Science ®Google Scholar
• Wang B. Random attractors for the stochastic FitzHugh–Nagumo system on unbounded domains. Nonlinear Anal TMA. 2009;71:2811–2828. doi: 10.1016/j.na.2009.01.131
   Web of Science ®Google Scholar
• Wang B. Upper semicontinuity of random attractors for non-compact random systems. J Differ Equ. 2009;139:1–18.
   Google Scholar
• Wang X, Lu K, Wang B. Wong-Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains. J Differ Equ. 2018;264:378–424. doi: 10.1016/j.jde.2017.09.006
   Web of Science ®Google Scholar
• Wang B. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differ Equ. 2012;253:1544–1583. doi: 10.1016/j.jde.2012.05.015
   Web of Science ®Google Scholar
• Wang B. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete Contin Dyn Syst Ser A. 2014;34:269–300. doi: 10.3934/dcds.2014.34.269
   Web of Science ®Google Scholar
• Wang B. Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch. Dyn. 2014;14(4):1–31. doi: 10.1142/S0219493714500099
   Web of Science ®Google Scholar
• Yin F, Liu L. D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains. Comput Math Appl. 2014;68:424–438. doi: 10.1016/j.camwa.2014.06.018
   Web of Science ®Google Scholar
• Zhou S, Tian Y, Wang W. Fractal dimension of random attractors for stochastic non-autonomous reaction–diffusion equations. Appl Math Comput. 2016;276:80–95.
   Web of Science ®Google Scholar
• Zhou S. Random exponential attractor for stochastic reaction–diffusion equation with multiplicative noise in R3. J Differ Equ. 2017;263:6347–6383. doi: 10.1016/j.jde.2017.07.013
   Web of Science ®Google Scholar
• Liu L, Caraballo T. Well-posedness and dynamics of a fractional stochastic integro-differential equation. Phys D. 2017;355:45–57. doi: 10.1016/j.physd.2017.05.006
   Web of Science ®Google Scholar
• Lu H, Lv S. Random attrator for fractional Ginzburg-Laudau equation with multiplicative noise. Taiwanese J Math. 2014;18(2):435–450. doi: 10.11650/tjm.18.2014.3053
   Web of Science ®Google Scholar
• Lu H, Bates PW, Xin J, et al. Asymptotic behavior of stochastic fractional power dissipative equations on Rn. Nonlinear Anal TMA. 2015;128:176–198. doi: 10.1016/j.na.2015.06.033
   Web of Science ®Google Scholar
• Lu H, Bates PW, Lu S, et al. Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain. Commmu Math Sci. 2016;14:273–295. doi: 10.4310/CMS.2016.v14.n1.a11
   Web of Science ®Google Scholar
• Shen T, Huang J. Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials. Nonlinear Anal TMA. 2014;110:33–46. doi: 10.1016/j.na.2014.06.018
   Web of Science ®Google Scholar
• Shu J, Li P, Zhang J, et al. Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise. J Math Phys. 2015;56:102702. doi: 10.1063/1.4934724
   Web of Science ®Google Scholar
• Gu A, Li D, Wang B, et al. Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn. J Differ Equ. 2018;264:7094–7137. doi: 10.1016/j.jde.2018.02.011
   Web of Science ®Google Scholar
• Coleman BD, Gurtin ME. Equipresence and constitutive equations for rigid heat conductors. Z Angew Math Phys. 1967;18:199–208. doi: 10.1007/BF01596912
   Web of Science ®Google Scholar
• Guo B, Zhou G. Ergodicity of the stochastic fractional reaction–diffusion equation. Nonlinear Anal. 2014;109:1–22. doi: 10.1016/j.na.2014.06.008
   Web of Science ®Google Scholar
• Caraballo T, Real J, Chueshov ID. Pullback attractors for stochastic heat equations in materials with memory. Discrete Contin Dyn Syst Ser B. 2008;9(3):525–539.
   Web of Science ®Google Scholar
• Caraballo T, Chueshov ID, Marin-Rubio P. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete Contin Dyn Syst. 2012;18(2–3):253–270.
   Google Scholar
• Conti M, Pata V, Squassina M. Singular limit of differential systems with memory. Indiana Univ Math J. 2006;55(1):169–215. doi: 10.1512/iumj.2006.55.2661
   Web of Science ®Google Scholar
• Giorgi C, Pata V, Marzocchi A. Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differ Equ Appl. 1998;5:333–354. doi: 10.1007/s000300050049
   Web of Science ®Google Scholar
• Grasselli M, Pata V. Uniform attractors of nonautonomous dynamical systems with memory. In: Progress in nonlinear differential equations and their applications. vol. 50. Basel: Birkhäuser Verlag; 2002, p. 155–178.
   Google Scholar
• DiNezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573. doi: 10.1016/j.bulsci.2011.12.004
   Web of Science ®Google Scholar
• Lu H, Bates PW, Lu S, et al. Dynamics of 3-D fractional complex Ginzburg–Landau equation. J Differ Equ. 2015;259:5276–5301. doi: 10.1016/j.jde.2015.06.028
   Web of Science ®Google Scholar
• Lions JL. Quelques methodes de resolution des problemes aux limites non lineaires. Paris: Dunod; 1969.
   Google Scholar