# American Institute of Mathematical Sciences

April  2022, 11(2): 439-455. doi: 10.3934/eect.2021007

## Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative

 1 Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam 2 Faculty of Mathematics and Computer Science, University of Science, VNUHCM Ho Chi Minh City, Vietnam 3 Department of Applied Mathematics, Bharathiar University, Coimbatore 641 046, India 4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Corresponding author: Nguyen Huy Tuan (nguyenhuytuan@tdmu.edu.vn)

Received  May 2020 Revised  December 2020 Published  April 2022 Early access  January 2021

Fund Project: This paper is supported by Thu Dau Mot University. The authors wish to express their sincere thanks to the referees and the editor for their valuable comments

In this paper, we consider a nonlinear fractional diffusion equations with a Riemann-Liouville derivative. First, we establish the global existence and uniqueness of mild solutions under some assumptions on the input data. Some regularity results for the mild solution and its derivatives of fractional orders are also derived. Our key idea is to combine the theories of Mittag-Leffler functions, Banach fixed point theorem and some Sobolev embeddings.

Citation: Tran Bao Ngoc, Nguyen Huy Tuan, R. Sakthivel, Donal O'Regan. Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative. Evolution Equations and Control Theory, 2022, 11 (2) : 439-455. doi: 10.3934/eect.2021007
##### References:
 [1] E. A. Abdel-Rehim, From power laws to fractional diffusion processes with and without external forces, the non direct way., Fract. Calc. Appl. Anal., 22 (2019), 60-77.  doi: 10.1515/fca-2019-0004. [2] M. Abramowitz and I. A. Stegun, Table Errata: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972. [3] E. Alvarez, G. Ciprian, V. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016. [4] L. Banjai and E. Otárola, A PDE approach to fractional diffusion: A space-fractional wave equation, Numer. Math., 143 (2019), 177-222.  doi: 10.1007/s00211-019-01055-5. [5] M. Benchohra, S. Bouriah and J. J. Nieto, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr. Math., 52 (2019), 437-450.  doi: 10.1515/dema-2019-0032. [6] G. Di Blasio, Time and space Sobolev regularity of solutions to homogeneous parabolic equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 89-94. [7] G. Di Blasio, Sobolev regularity for solutions of parabolic equations by extrapolation methods, Adv. Differential Equations, 6 (2001), 481-512. [8] G. Di Blasio, Maximal $L^p$ regularity for nonautonomous parabolic equations in extrapolation spaces, J. Evol. Equ., 6 (2006), 229-245.  doi: 10.1007/s00028-006-0241-3. [9] M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725. [10] L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004. [11] Y. Chen, H. Gao, M. Garrido-Atienza and B. Schmalfuss, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79. [12] F. Colombo and D. Guidetti, A unified approach to nonlinear integro-differential inverse problems of parabolic type, Z. Anal. Anwendungen, 21 (2002), 431-464.  doi: 10.4171/ZAA/1086. [13] C. G. Gal and M. Warma, Fractional in-Time Semilinear Parabolic Equations and Applications, Sprinter International Publishing, 2020,184 pp. doi: 10.1007/978-3-030-45043-4. [14] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [15] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time fractional diffusion: A discrete random walk approach,, Nonlinear. Dynam., 29 (2002), 129-143.  doi: 10.1023/A:1016547232119. [16] A. Khan, H. Khan, J. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equation with Mittag-Leffler kernel, Chaos Solitons Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026. [17] H. Khan, J. F. Gómez-Aguilar, A. Khan and T. S. Khan, Stability analysis for fractional order advection-reaction diffusion system, Phys. A, 521 (2019), 737-751.  doi: 10.1016/j.physa.2019.01.102. [18] Y. Kian, L. Oksanen, E. Soccorsi and M. Yamamoto, Global uniqueness in an inverse problem for time fractional diffusion equations, J. Differential Equations, 264 (2018), 1146-1170.  doi: 10.1016/j.jde.2017.09.032. [19] B. Li and X. Xie, Regularity of solutions to time fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3195-3210.  doi: 10.3934/dcdsb.2018340. [20] M. Magdziarz, R. Metzler, W. Szczotka and P. Zebrowski, doititleCorrelated continuous-time random walks in external force fields, Phys. Rev. E, 85 (2012), 051103. doi: 10.1103/PhysRevE.85.051103. [21] F. Mainardi, Fractional diffusive waves in viscoelastic solids Nonlinear Waves in Solids, Fairfield, NJ, ASME/AMR, 93–97. [22] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach., Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3. [23] V. F. Morales-Delgado, J. F. Gómez-Aguilar, Khaled M. Saad, M. A. Khan and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach., Phys. A, 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018. [24] R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150. [25] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x. [26] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, Inc., San Diego, CA, 1999. [27] K. Sakamoto and 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.  doi: 10.1016/j.jmaa.2011.04.058. [28] T. Sandev, R. Metzler and V. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative., J. Phys. A, 44 (2011), 255203, 21 pp. doi: 10.1088/1751-8113/44/25/255203. [29] H. Ye, J. Gao and Y. Ding, A generalized Grönwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061. [30] S. Zhang, Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal., 71 (2009), 2087–2093. doi: 10.1016/j.na.2009.01.043.

show all references

##### References:
 [1] E. A. Abdel-Rehim, From power laws to fractional diffusion processes with and without external forces, the non direct way., Fract. Calc. Appl. Anal., 22 (2019), 60-77.  doi: 10.1515/fca-2019-0004. [2] M. Abramowitz and I. A. Stegun, Table Errata: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972. [3] E. Alvarez, G. Ciprian, V. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016. [4] L. Banjai and E. Otárola, A PDE approach to fractional diffusion: A space-fractional wave equation, Numer. Math., 143 (2019), 177-222.  doi: 10.1007/s00211-019-01055-5. [5] M. Benchohra, S. Bouriah and J. J. Nieto, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr. Math., 52 (2019), 437-450.  doi: 10.1515/dema-2019-0032. [6] G. Di Blasio, Time and space Sobolev regularity of solutions to homogeneous parabolic equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (1998), 89-94. [7] G. Di Blasio, Sobolev regularity for solutions of parabolic equations by extrapolation methods, Adv. Differential Equations, 6 (2001), 481-512. [8] G. Di Blasio, Maximal $L^p$ regularity for nonautonomous parabolic equations in extrapolation spaces, J. Evol. Equ., 6 (2006), 229-245.  doi: 10.1007/s00028-006-0241-3. [9] M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725. [10] L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004. [11] Y. Chen, H. Gao, M. Garrido-Atienza and B. Schmalfuss, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79. [12] F. Colombo and D. Guidetti, A unified approach to nonlinear integro-differential inverse problems of parabolic type, Z. Anal. Anwendungen, 21 (2002), 431-464.  doi: 10.4171/ZAA/1086. [13] C. G. Gal and M. Warma, Fractional in-Time Semilinear Parabolic Equations and Applications, Sprinter International Publishing, 2020,184 pp. doi: 10.1007/978-3-030-45043-4. [14] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [15] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time fractional diffusion: A discrete random walk approach,, Nonlinear. Dynam., 29 (2002), 129-143.  doi: 10.1023/A:1016547232119. [16] A. Khan, H. Khan, J. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equation with Mittag-Leffler kernel, Chaos Solitons Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026. [17] H. Khan, J. F. Gómez-Aguilar, A. Khan and T. S. Khan, Stability analysis for fractional order advection-reaction diffusion system, Phys. A, 521 (2019), 737-751.  doi: 10.1016/j.physa.2019.01.102. [18] Y. Kian, L. Oksanen, E. Soccorsi and M. Yamamoto, Global uniqueness in an inverse problem for time fractional diffusion equations, J. Differential Equations, 264 (2018), 1146-1170.  doi: 10.1016/j.jde.2017.09.032. [19] B. Li and X. Xie, Regularity of solutions to time fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3195-3210.  doi: 10.3934/dcdsb.2018340. [20] M. Magdziarz, R. Metzler, W. Szczotka and P. Zebrowski, doititleCorrelated continuous-time random walks in external force fields, Phys. Rev. E, 85 (2012), 051103. doi: 10.1103/PhysRevE.85.051103. [21] F. Mainardi, Fractional diffusive waves in viscoelastic solids Nonlinear Waves in Solids, Fairfield, NJ, ASME/AMR, 93–97. [22] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach., Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3. [23] V. F. Morales-Delgado, J. F. Gómez-Aguilar, Khaled M. Saad, M. A. Khan and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach., Phys. A, 523 (2019), 48-65.  doi: 10.1016/j.physa.2019.02.018. [24] R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150. [25] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x. [26] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, Inc., San Diego, CA, 1999. [27] K. Sakamoto and 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.  doi: 10.1016/j.jmaa.2011.04.058. [28] T. Sandev, R. Metzler and V. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative., J. Phys. A, 44 (2011), 255203, 21 pp. doi: 10.1088/1751-8113/44/25/255203. [29] H. Ye, J. Gao and Y. Ding, A generalized Grönwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061. [30] S. Zhang, Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal., 71 (2009), 2087–2093. doi: 10.1016/j.na.2009.01.043.
 [1] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [2] Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control and Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016 [3] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [4] María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions. Electronic Research Archive, 2020, 28 (2) : 567-587. doi: 10.3934/era.2020030 [5] Paul Eloe, Jaganmohan Jonnalagadda. Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2719-2734. doi: 10.3934/dcdss.2020220 [6] Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837 [7] Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 [8] Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021026 [9] Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 [10] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 [11] Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3195-3210. doi: 10.3934/dcdsb.2018340 [12] Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321 [13] Xinchi Huang, Atsushi Kawamoto. Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates. Inverse Problems and Imaging, 2022, 16 (1) : 39-67. doi: 10.3934/ipi.2021040 [14] Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic and Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006 [15] Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure and Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605 [16] Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435 [17] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [18] Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122 [19] Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173 [20] Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

2020 Impact Factor: 1.081