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18/12/2025
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Nguyen, V. D. (2025). Existence result for inverse problems governed by generalized rayleigh-tokes equations. Journal of Water Resources & Environmental Engineering, 97, 72-78. https://dev.vojs.vn/index.php/jwree/article/view/17
Existence result for inverse problems governed by generalized rayleigh-tokes equations
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Abstract
In this article, we study an inverse source problem related to the generalized Rayleigh–Stokes equations on Hilbert scales, with weak-valued nonlinearities and memory effects depending on the history of the state function. We establish a representation of the mild solution and then investigate crucial properties of the associated resolvent operators. Based on these analyses, the existence result is obtained by applying Banach’s fixed-point principle. Finally, an illustrative example is provided to demonstrate the theoretical results.
Article Details
Keywords
Rayleigh-Stokes equations, weak nonlinearity, inverse problems, existence results
References
E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, (2015), An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131, 1-31.
C.-M. Chen, F. Liu, K. Burrage and Y. Chen, (2013), Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78, 924-944.
P. Clément, J. A. Nohel, (1981), Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12, 514–535.
C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, (2009), The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60, 921-933.
N. V. Dac, T. D. Ke and L. T. P. Thuy, (2023), On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities, D.C.D.S, 16, 2883-2901.
N. V. Dac, T. D. Ke, V. N. Phong, (2025), On stability and regularity of solutions to generalized Rayleigh-Stokes equations involving delays in Hilbert scales, E.E.C.T, 14(2), 289-312.
J. Janno and K. Kasemets, (2017), Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Probl., 25, 777-798.
M. Khan, (2009), The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10, 3190-3195.
D. Lan and P. T. Tuan, (2022), On stability for semilinear generalized Rayleigh–Stokes equation involving delays, Quart. Appl. Math., 80, 701-715.
D. V. Loi and T. V. Tuan, (2023), Stability analysis for a class of semilinear nonlocal evolution equations, Bol. Soc. Mat. Mex., 29, Paper No. 46, 22 pp.
T. V. Tuan, (2023), Stability and regularity in inverse source problem for generalized subdiffusion equation perturbed by locally Lipschitz sources, Z. Angew. Math. Phys., 74, no. 2, Paper No. 65.
N. H. Tuan, Y. Zhou, L. D. Long and N. H. Can, (2020), Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative, Comput. Appl. Math., 39, No. 75, 16 pp.
T. D. Ke, L. T. P. Thuy and P. T. Tuan, (2022), An inverse source problem for generalized Rayleigh-Stokes equations involving superlinear perturbations, J. Math. Anal. Appl., 507(2), 125797.
V. Vergara and R. Zacher, (2015), Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47, 210–239.
C.-M. Chen, F. Liu, K. Burrage and Y. Chen, (2013), Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78, 924-944.
P. Clément, J. A. Nohel, (1981), Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12, 514–535.
C. Fetecau, M. Jamil, C. Fetecau and D. Vieru, (2009), The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys., 60, 921-933.
N. V. Dac, T. D. Ke and L. T. P. Thuy, (2023), On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities, D.C.D.S, 16, 2883-2901.
N. V. Dac, T. D. Ke, V. N. Phong, (2025), On stability and regularity of solutions to generalized Rayleigh-Stokes equations involving delays in Hilbert scales, E.E.C.T, 14(2), 289-312.
J. Janno and K. Kasemets, (2017), Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Probl., 25, 777-798.
M. Khan, (2009), The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl., 10, 3190-3195.
D. Lan and P. T. Tuan, (2022), On stability for semilinear generalized Rayleigh–Stokes equation involving delays, Quart. Appl. Math., 80, 701-715.
D. V. Loi and T. V. Tuan, (2023), Stability analysis for a class of semilinear nonlocal evolution equations, Bol. Soc. Mat. Mex., 29, Paper No. 46, 22 pp.
T. V. Tuan, (2023), Stability and regularity in inverse source problem for generalized subdiffusion equation perturbed by locally Lipschitz sources, Z. Angew. Math. Phys., 74, no. 2, Paper No. 65.
N. H. Tuan, Y. Zhou, L. D. Long and N. H. Can, (2020), Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative, Comput. Appl. Math., 39, No. 75, 16 pp.
T. D. Ke, L. T. P. Thuy and P. T. Tuan, (2022), An inverse source problem for generalized Rayleigh-Stokes equations involving superlinear perturbations, J. Math. Anal. Appl., 507(2), 125797.
V. Vergara and R. Zacher, (2015), Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47, 210–239.