We consider a functional stochastic delay semilinear Rayleigh–Stokes equation involving Riemann–Liouville derivative. Our aim is using the resolvent theory, fixed point argument to prove the global solvability and gives some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment.
AlosE.MazetO.NualartD., Stochastic calculus with respect to Gaussian processes, The Annals of Probability29(2) (2001), 766–801.
2.
BazhlekovaE.JinB.LazarovR.ZhouZ., An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid, Numerische Mathematik131(1) (2015), 1–31. doi:10.1007/s00211-014-0685-2.
3.
BiX.MuS.LiuQ.LiuQ.LiuB.ZhuangP.GaoJ.JiangH.LiX.LiB., Advanced implicit meshless approaches for the Rayleigh–Stokes–Stokes problem for a heated generalized second grade fluid with fractional derivative, International Journal of Computational Methods15(05) (2018), 1850032. doi:10.1142/S0219876218500329.
4.
BonforteM.SireY.VázquezJ.L., Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, 2014, arXiv preprint. arXiv:1404.6195.
5.
BoufoussiB.HajjiS., Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics & probability letters82(8) (2012), 1549–1558. doi:10.1016/j.spl.2012.04.013.
6.
BoufoussiB.HajjiS.LakhelE.H., Functional differential equations in Hilbert spaces driven by a fractional Brownian motion, Afrika Matematika23 (2012), 173–194. doi:10.1007/s13370-011-0028-8.
7.
CaraballoT.Garrido-A.M.J.TaniguchiT., The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods & Applications74(11) (2011), 3671–3684. doi:10.1016/j.na.2011.02.047.
8.
CarmiS.TurgemanL.BarkaiE., On distributions of functionals of anomalous diffusion paths, Journal of Statistical Physics141 (2010), 1071–1092. doi:10.1007/s10955-010-0086-6.
9.
ChenC.M.LiuF.AnhV., Numerical analysis of the Rayleigh–Stokes–Stokes problem for a heated generalized second grade fluid with fractional derivatives, Applied Mathematics and Computation204(1) (2008), 340–351. doi:10.1016/j.amc.2008.06.052.
10.
ChenC.M.LiuF.BurrageK.ChenY., Numerical methods of the variable-order Rayleigh–Stokes–Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA Journal of Applied Mathematics78(5) (2013), 924–944. doi:10.1093/imamat/hxr079.
11.
ChoudharyA.KumarD.SinghJ., Analytical solution of fractional differential equations arising in fluid mechanics by using Sumudu transform method, Nonlinear Engineering3(3) (2014), 133–139. doi:10.1515/nleng-2014-0007.
12.
DebnathL., Fractional integrals and fractional differential equations in fluid mechanics, Fract. Calc. Appl. Anal.6, 119–155.
13.
DungN.T., Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays, Journal of the Korean Statistical Society43(4) (2014), 599–608. doi:10.1016/j.jkss.2014.02.003.
14.
FerranteM.RoviraC., Convergence of delay differential equations driven by fractional Brownian motion, Journal of Evolution Equations10 (2010), 761–783. doi:10.1007/s00028-010-0069-8.
15.
FetecauC.JamilM.FetecauC.VieruD., The Rayleigh–Stokes–Stokes problem for an edge in a generalized Oldroyd-b fluid, Zeitschrift für angewandte Mathematik und Physik60 (2009), 921–933. doi:10.1007/s00033-008-8055-5.
16.
GrilloG.MuratoriM.PunzoF., On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst.35 (2015), 5927–5962. doi:10.3934/dcds.2015.35.5927.
17.
KhanM., The Rayleigh–Stokes–Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Analysis: Real World Applications10(5) (2009), 3190–3195.
18.
LanD., Regularity and stability analysis for semilinear generalized Rayleigh–Stokes equations, Evolution Equations & Control Theory11(1) (2022).
19.
LucN.H.TuanN.H.ZhouY., Regularity of the solution for a final value problem for the Rayleigh–Stokes equation, Mathematical Methods in the Applied Sciences42(10) (2019), 3481–3495. doi:10.1002/mma.5593.
20.
NgocT.B.LucN.H.AuV.V.TuanN.H.ZhouY., Existence and regularity of inverse problem for the nonlinear fractional Rayleigh–Stokes equations, Mathematical Methods in the Applied Sciences44(3) (2021), 2532–2558. doi:10.1002/mma.6162.
21.
NualartD., The Malliavin Calculus and Related Topics, Vol. 1995, Springer, 2006.
22.
QuanN.N., Existence and asymptotic stability for lattice stochastic integrodifferential equations with infinite delays, Journal of Integral Equations and Applications34(3) (2022), 357–372.
23.
RossikhinY.A.ShitikovaM.V., Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica120(1–4) (1997), 109–125. doi:10.1007/BF01174319.
24.
SakthivelR.RevathiP.RenY., Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Analysis: Theory, Methods & Applications81 (2013), 70–86. doi:10.1016/j.na.2012.10.009.
25.
SalehiF.SaeediH.MoghadamM.M., Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh–Stokes problem, Computational and Applied Mathematics37 (2018), 5274–5292. doi:10.1007/s40314-018-0631-5.
26.
ShenF.TanW.ZhaoY.MasuokaT., The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Analysis: Real World Applications7(5) (2006), 1072–1080.
27.
TindelS.TudorC.A.ViensF., Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields127 (2003), 186–204. doi:10.1007/s00440-003-0282-2.
28.
TuanN.H.ZhouY.ThachT.N.CanN.H., Initial inverse problem for the nonlinear fractional Rayleigh–Stokes equation with random discrete data, Communications in Nonlinear Science and Numerical Simulation78 (2019), 104873. doi:10.1016/j.cnsns.2019.104873.
29.
XueC.NieJ., Exact solutions of the Rayleigh–Stokes problem for a heated generalized second grade fluid in a porous half-space, Applied Mathematical Modelling33(1) (2009), 524–531. doi:10.1016/j.apm.2007.11.015.
30.
ZakyM.A., An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid, Computers & mathematics with Applications75(7) (2018), 2243–2258. doi:10.1016/j.camwa.2017.12.004.
31.
ZierepJ.BohningR.FetecauC., Rayleigh–Stokes problem for non-Newtonian medium with memory, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics87(6) (2007), 462–467. doi:10.1002/zamm.200710328.
