In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.
Citation: Özge Arıbaş, İsmet Gölgeleyen, Mustafa Yıldız. On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation[J]. AIMS Mathematics, 2023, 8(3): 5432-5444. doi: 10.3934/math.2023273
In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.
| [1] |
J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 25 (2009), 1–16. https://doi.org/10.1088/0266-5611/25/11/115002 doi: 10.1088/0266-5611/25/11/115002
|
| [2] |
Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59 (2010), 1766–1772. https://doi.org/10.1016/j.camwa.2009.08.015 doi: 10.1016/j.camwa.2009.08.015
|
| [3] |
K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426–447. https://doi.org/10.1016/j.jmaa.2011.04.058 doi: 10.1016/j.jmaa.2011.04.058
|
| [4] |
L. Li, L. Y. Jin, S. M. Fang, Existence and uniqueness of the solution to a coupled fractional diffusion system, Adv. Differ. Equ., 370 (2015), 1–14. https://doi.org/10.1186/s13662-015-0707-0 doi: 10.1186/s13662-015-0707-0
|
| [5] |
Y. Luchko, W. Rundell, M. Yamamoto, L. H. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Probl., 29 (2013), 1–16. https://doi.org/10.1088/0266-5611/29/6/065019 doi: 10.1088/0266-5611/29/6/065019
|
| [6] |
B. T. Jin, B. Y. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. https://doi.org/10.1137/16M1089320 doi: 10.1137/16M1089320
|
| [7] | B. T. Jin, Fractional differential equations, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-76043-4 |
| [8] |
Y. Kian, M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. App. Anal., 20 (2017), 117–138. https://doi.org/10.1515/fca-2017-0006 doi: 10.1515/fca-2017-0006
|
| [9] |
T. B. Ngoc, N. H. Tuan, D. O'Regan, Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104882. https://doi.org/10.1016/j.cnsns.2019.104882 doi: 10.1016/j.cnsns.2019.104882
|
| [10] |
M. H. Tiwana, K. Maqbool, A. B. Mann, Homotopy perturbation Laplace transform solution of fractional non-linear reaction diffusion system of Lotka-Volterra type differential equation, Eng. Sci. Technol. Int. J., 20 (2017), 672–678. https://doi.org/10.1016/j.jestch.2016.10.014 doi: 10.1016/j.jestch.2016.10.014
|
| [11] |
W. L. Qiu, D. Xu, J. Guo, J. Zhou, A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model, Numer. Algorithms, 85 (2020), 39–58. https://doi.org/10.1007/s11075-019-00801-y doi: 10.1007/s11075-019-00801-y
|
| [12] |
W. L. Qiu, X. Xiao, K. X. Li, Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity, Commun. Nonlinear Sci. Numer. Simul., 116 (2023), 106804. https://doi.org/10.1016/j.cnsns.2022.106804 doi: 10.1016/j.cnsns.2022.106804
|
| [13] | V. S. Vladimirov, Equations of mathematical physics, New York: Marcel Dekker, 1971. |
| [14] | E. U. Tarafdar, M. S. R. Chowhury, Topological methods for set-valued nonlinear analysis, Singapore: World Scientific, 2008. https://doi.org/10.1142/6347 |
| [15] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006. |
| [16] | I. Podlubny, Fractional differential equations, Academic Press, 1998. |
| [17] | R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., Elsevier, 2003. |
| [18] | H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-70914-7 |
| [19] | L. C. Evans, Partial differential equations, Providence, RI: American Mathematical Society, 1998. |
| [20] | A. Henrot, Extremum problems for eigenvalues of elliptic operators, Basel, Switzerland: Birkhauser, 2006. |
| [21] | J. K. Hunter, B. Nachtergaele, Applied analysis, Singapore: World Scientific, 2001. |
Özge Arıbaş, İsmet Gölgeleyen, Mustafa Yıldız. On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation[J]. AIMS Mathematics, 2023, 8(3): 5432-5444. doi: 10.3934/math.2023273
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