# American Institute of Mathematical Sciences

December  2021, 26(12): 6483-6510. doi: 10.3934/dcdsb.2021030

## Initial value problem for fractional Volterra integro-differential equations with Caputo derivative

 1 Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam 2 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China 4 Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China 5 Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

* Corresponding author: Huu Can Nguyen (nguyenhuucan@tdtu.edu.vn)

Dedicated to Tomás Caraballo on his 60th birthday.

Received  April 2020 Revised  December 2020 Published  December 2021 Early access  February 2021

In this paper, we consider the time-fractional Volterra integro-differential equations with Caputo derivative. For globally Lispchitz source term, we investigate the global existence for a mild solution. The main tool is to apply the Banach fixed point theorem on some new weighted spaces combining some techniques on the Wright functions. For the locally Lipschitz case, we study the existence of local mild solutions to the problem and provide a blow-up alternative for mild solutions. We also establish the problem of continuous dependence with respect to initial data. Finally, we present some examples to illustrate the theoretical results.

Citation: Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030
##### References:
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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.  Google Scholar [6] 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.  Google Scholar [7] T. Caraballo and J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [8] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar [9] 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 and Continuous Dynamical Systems - Series A, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.  Google Scholar [10] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar [11] B. D. Coleman and V. J. Mizel, Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal., 28 (1966), 87-123.  doi: 10.1007/BF00251727.  Google Scholar [12] M. Conti, E. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.  doi: 10.1016/j.na.2013.08.015.  Google Scholar [13] M. Conti, F. Dell'Oro and V. 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Simul., 42 (2017), 204-214.  doi: 10.1016/j.cnsns.2016.05.021.  Google Scholar [18] M. L. Heard and S. M. Rankin III, A semilinear parabolic Volterra integro-differential equation, J. Differential Equations, 71 (1988), 201-233.  doi: 10.1016/0022-0396(88)90023-X.  Google Scholar [19] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779.  Google Scholar [20] L. Li and G. J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.  Google Scholar [21] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.  Google Scholar [22] R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math., 35 (1977), 1-19.  doi: 10.1090/qam/452184.  Google Scholar [23] F. Mainardi, A. Mura and G. Pagnini, The $M$-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ., (2010), 104505, 29 pp. doi: 10.1155/2010/104505.  Google Scholar [24] L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984.  doi: 10.1002/mma.4779.  Google Scholar [25] I. Podlubny, Fractional Differential Equations, Academic press, California, 1999.  doi: 10.1016/978-0-12-558840-9.  Google Scholar [26] M. H. M. Rashid and Y. E. Qaderi, Semilinear fractional integro-differential equations with compact semigroup, Nonlinear Anal., 71 (2009), 6276-6282.  doi: 10.1016/j.na.2009.06.035.  Google Scholar [27] M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503.  doi: 10.1016/j.cnsns.2010.12.043.  Google Scholar [28] 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.  Google Scholar [29] J. V. C. Sousa, F. G. Rodrigues and E. C. Oliveira, Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033-3043.  doi: 10.1002/mma.5563.  Google Scholar [30] H. G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1515/fca-2020-0012.  Google Scholar [31] A. Viana, Local well-posedness for a Lotka-Volterra system in Besov spaces, Comput. Math. Appl., 69 (2015), 667-674.  doi: 10.1016/j.camwa.2015.02.013.  Google Scholar [32] A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335.  Google Scholar [33] G. Webb, An abstract semilinear Volterra integro-differential equation, Proc. Amer. Math. Soc., 69 (1978), 255-260.  doi: 10.1090/S0002-9939-1978-0467214-4.  Google Scholar [34] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Different. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar [35] Y. Zhou, J. Manimaran, L. Shangerganesh and A. Debbouche, Weakness and Mittag-Leffler stability of solutions for time-fractional Keller-Segel models, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 753-761.  doi: 10.1515/ijnsns-2018-0035.  Google Scholar

show all references

##### References:
 [1] B. Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.  Google Scholar [2] B. Andrade and A. Viana, Integrodifferential equations with applications to a plate equation with memory, Mathematische Nachrichten, 289 (17–18), 2159-2172.  doi: 10.1002/mana.201500205.  Google Scholar [3] B. Andrade, A. N. Carvalho, P. M. Carvalho-Neto and P. Marin-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topological Methods in Nonlinear Analysis, 45 (2015), 439-467.  doi: 10.12775/TMNA.2015.022.  Google Scholar [4] K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal., 72 (2010), 4587-4593.  doi: 10.1016/j.na.2010.02.035.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] T. Caraballo and J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differ. Equ., 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [8] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar [9] 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 and Continuous Dynamical Systems - Series A, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.  Google Scholar [10] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors,, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.  Google Scholar [11] B. D. Coleman and V. J. Mizel, Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal., 28 (1966), 87-123.  doi: 10.1007/BF00251727.  Google Scholar [12] M. Conti, E. Marchini and V. Pata, A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.  doi: 10.1016/j.na.2013.08.015.  Google Scholar [13] M. Conti, F. Dell'Oro and V. Pata, Nonclassical diffusion with memory lacking instantaneous damping, Communications on Pure and Applied Analysis, 19 (2020). doi: 10.3934/cpaa.2020090.  Google Scholar [14] M. Conti, E. Marchini and V. Pata, Reaction-diffusion with memory in the minimal state framework, Trans. Amer. Math. Soc., 366 (2014), 4969-4986.  doi: 10.1090/S0002-9947-2013-06097-7.  Google Scholar [15] M. D'Abbico, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal, 95 (2014), 130-145.  doi: 10.1016/j.na.2013.09.006.  Google Scholar [16] M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Asymp. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.  Google Scholar [17] H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214.  doi: 10.1016/j.cnsns.2016.05.021.  Google Scholar [18] M. L. Heard and S. M. Rankin III, A semilinear parabolic Volterra integro-differential equation, J. Differential Equations, 71 (1988), 201-233.  doi: 10.1016/0022-0396(88)90023-X.  Google Scholar [19] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779.  Google Scholar [20] L. Li and G. J. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.  Google Scholar [21] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.  Google Scholar [22] R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math., 35 (1977), 1-19.  doi: 10.1090/qam/452184.  Google Scholar [23] F. Mainardi, A. Mura and G. Pagnini, The $M$-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ., (2010), 104505, 29 pp. doi: 10.1155/2010/104505.  Google Scholar [24] L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984.  doi: 10.1002/mma.4779.  Google Scholar [25] I. Podlubny, Fractional Differential Equations, Academic press, California, 1999.  doi: 10.1016/978-0-12-558840-9.  Google Scholar [26] M. H. M. Rashid and Y. E. Qaderi, Semilinear fractional integro-differential equations with compact semigroup, Nonlinear Anal., 71 (2009), 6276-6282.  doi: 10.1016/j.na.2009.06.035.  Google Scholar [27] M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503.  doi: 10.1016/j.cnsns.2010.12.043.  Google Scholar [28] 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.  Google Scholar [29] J. V. C. Sousa, F. G. Rodrigues and E. C. Oliveira, Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, Math. Methods Appl. Sci., 42 (2019), 3033-3043.  doi: 10.1002/mma.5563.  Google Scholar [30] H. G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231.  doi: 10.1515/fca-2020-0012.  Google Scholar [31] A. Viana, Local well-posedness for a Lotka-Volterra system in Besov spaces, Comput. Math. Appl., 69 (2015), 667-674.  doi: 10.1016/j.camwa.2015.02.013.  Google Scholar [32] A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335.  Google Scholar [33] G. Webb, An abstract semilinear Volterra integro-differential equation, Proc. Amer. Math. Soc., 69 (1978), 255-260.  doi: 10.1090/S0002-9939-1978-0467214-4.  Google Scholar [34] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Different. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar [35] Y. Zhou, J. Manimaran, L. Shangerganesh and A. Debbouche, Weakness and Mittag-Leffler stability of solutions for time-fractional Keller-Segel models, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 753-761.  doi: 10.1515/ijnsns-2018-0035.  Google Scholar
Ex1. The solutions $u(x,t)$ at $t \in \{0.1, 0.5, 0.9\}$ for $\alpha = 0.2$ and $\epsilon \in \{0.1, 0.01, 0.001\}$
Ex1. The solutions $u(x,t)$ on $(x,t) \in (0,\pi) \times (0,1)$ for $\alpha = 0.2$ and $\epsilon \in \{0.1, 0.01, 0.001\}$
Ex2. The solutions $u(x,t)$ at $t \in \{0.1, 0.5, 0.9\}$ for $\alpha = 0.9$ and $\epsilon \in \{0.1, 0.01, 0.001\}$
Ex2. The solutions $u(x,t)$ on $(x,t) \in (0,\pi) \times (0,1)$ for $\alpha = 0.9$ and $\epsilon \in \{0.1, 0.01, 0.001\}$
Ex1. The error estimation for $\alpha = 0.2$ and $t \in \{0.1, 0.5, 0.9\}$
 $\epsilon = |\alpha^* - \alpha|$ $N(j) = 10,\; P =10,\; M = N =50$ Calculative error Percent error $\delta^{ \alpha'}_ \alpha$ $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.1) 0.1 0.011036875514009 15.46 % 0.01 0.004874547920700 6.83 % 0.001 0.002176754401323 3.05 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.5) 0.1 0.051353021961229 14.38 % 0.01 0.027169280169359 7.61 % 0.001 0.012084128051511 3.38 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.9) 0.1 0.074877070037197 11.65 % 0.01 0.042748723217464 6.65 % 0.001 0.028729381706881 4.47 %
 $\epsilon = |\alpha^* - \alpha|$ $N(j) = 10,\; P =10,\; M = N =50$ Calculative error Percent error $\delta^{ \alpha'}_ \alpha$ $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.1) 0.1 0.011036875514009 15.46 % 0.01 0.004874547920700 6.83 % 0.001 0.002176754401323 3.05 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.5) 0.1 0.051353021961229 14.38 % 0.01 0.027169280169359 7.61 % 0.001 0.012084128051511 3.38 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.9) 0.1 0.074877070037197 11.65 % 0.01 0.042748723217464 6.65 % 0.001 0.028729381706881 4.47 %
Ex2. The error estimation for $\alpha = 0.9$ and $t \in \{0.1, 0.5, 0.9\}$
 $\epsilon = |\alpha^* - \alpha|$ $N(j) = 10,\; P =10,\; M = N =50$ Calculative error Percent error $\delta^{ \alpha'}_ \alpha$ $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.1) 0.1 0.601407040658915 83.81 % 0.01 0.143648723675101 20.02 % 0.001 0.027371563705550 3.81 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.5) 0.1 0.643179867112674 79.89 % 0.01 0.167904683423887 20.85 % 0.001 0.025589340748129 3.18 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.9) 0.1 0.629790962460797 61.54 % 0.01 0.219601251416672 21.46 % 0.001 0.040291675260527 3.94 %
 $\epsilon = |\alpha^* - \alpha|$ $N(j) = 10,\; P =10,\; M = N =50$ Calculative error Percent error $\delta^{ \alpha'}_ \alpha$ $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.1) 0.1 0.601407040658915 83.81 % 0.01 0.143648723675101 20.02 % 0.001 0.027371563705550 3.81 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.5) 0.1 0.643179867112674 79.89 % 0.01 0.167904683423887 20.85 % 0.001 0.025589340748129 3.18 % $\mathrm{Error}^{ \alpha^*}_ \alpha$(0.9) 0.1 0.629790962460797 61.54 % 0.01 0.219601251416672 21.46 % 0.001 0.040291675260527 3.94 %
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