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February  2022, 11(1): 259-282. doi: 10.3934/eect.2021002

Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations

Do Lan

Faculty of Computer Science and Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, 100000, Vietnam

Received  June 2020 Revised  October 2020 Published  February 2022 Early access  January 2021

Fund Project: This research is funded by Thuyloi University Foundation for Science and Technology under grant number TLU.STF.19-04

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

Keywords: Rayleigh-Stokes problem, stability, nonlocal PDE.
Mathematics Subject Classification: Primary: 35B40, 35R11, 35C15; Secondary: 45D05, 45K05.
Citation: Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations and Control Theory, 2022, 11 (1) : 259-282. doi: 10.3934/eect.2021002
References:
[1]

E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.

[2]

X. Bi, S. Mu, Q. Liu, Q. Liu, B. Liu, P. Zhuang, J. Gao, H. Jiang, X. Li and B. Li, Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, Int. J. Comput. Methods, 15 (2018), 1850032, 27 pp. doi: 10.1142/S0219876218500329.

[3]

C. M. Chen, F. Liu, K. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.

[4]

C. M. Chen, F. Liu and V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput., 204 (2008), 340-351.  doi: 10.1016/j.amc.2008.06.052.

[5]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-0348-0387-8.

[6]

L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[7]

C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60 (2009), 921-933.  doi: 10.1007/s00033-008-8055-5.

[8]

M. Khan, The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10 (2009), 3190-3195.  doi: 10.1016/j.nonrwa.2008.10.002.

[9]

N. H. Luc, N. H. Tuan and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.

[10]

T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., (2020), 1–27. doi: 10.1002/mma.6162.

[11]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[12]

F. Salehi, H. Saeedi and M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274-5292.  doi: 10.1007/s40314-018-0631-5.

[13]

F. Shen, W. Tan, Y. Zhao and Y. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Anal. Real World Appl., 7 (2006), 1072-1080.  doi: 10.1016/j.nonrwa.2005.09.007.

[14]

N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104873, 18 pp. doi: 10.1016/j.cnsns.2019.104873.

[15]

C. Xue and J. Nie, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, Appl. Math. Model., 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015.

[16]

M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243-2258.  doi: 10.1016/j.camwa.2017.12.004.

[17]

J. Zierep, R. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, ZAMM Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.

[18]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Method Appl. Sci., (2019), 1–8. doi: 10.1002/mma.5926.

show all references

References:
[1]

E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.

[2]

X. Bi, S. Mu, Q. Liu, Q. Liu, B. Liu, P. Zhuang, J. Gao, H. Jiang, X. Li and B. Li, Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, Int. J. Comput. Methods, 15 (2018), 1850032, 27 pp. doi: 10.1142/S0219876218500329.

[3]

C. M. Chen, F. Liu, K. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.

[4]

C. M. Chen, F. Liu and V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput., 204 (2008), 340-351.  doi: 10.1016/j.amc.2008.06.052.

[5]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-0348-0387-8.

[6]

L. C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[7]

C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60 (2009), 921-933.  doi: 10.1007/s00033-008-8055-5.

[8]

M. Khan, The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10 (2009), 3190-3195.  doi: 10.1016/j.nonrwa.2008.10.002.

[9]

N. H. Luc, N. H. Tuan and Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci., 42 (2019), 3481-3495.  doi: 10.1002/mma.5593.

[10]

T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci., (2020), 1–27. doi: 10.1002/mma.6162.

[11]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[12]

F. Salehi, H. Saeedi and M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274-5292.  doi: 10.1007/s40314-018-0631-5.

[13]

F. Shen, W. Tan, Y. Zhao and Y. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Anal. Real World Appl., 7 (2006), 1072-1080.  doi: 10.1016/j.nonrwa.2005.09.007.

[14]

N. H. Tuan, Y. Zhou, T. N. Thach and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104873, 18 pp. doi: 10.1016/j.cnsns.2019.104873.

[15]

C. Xue and J. Nie, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, Appl. Math. Model., 33 (2009), 524-531.  doi: 10.1016/j.apm.2007.11.015.

[16]

M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243-2258.  doi: 10.1016/j.camwa.2017.12.004.

[17]

J. Zierep, R. Bohning and C. Fetecau, Rayleigh-Stokes problem for non-Newtonian medium with memory, ZAMM Z. Angew. Math. Mech., 87 (2007), 462-467.  doi: 10.1002/zamm.200710328.

[18]

Y. Zhou and J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Method Appl. Sci., (2019), 1–8. doi: 10.1002/mma.5926.

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