# 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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779. [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. [21] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066. [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. [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. [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. [25] I. Podlubny, Fractional Differential Equations, Academic press, California, 1999.  doi: 10.1016/978-0-12-558840-9. [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. [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. [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. [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. [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. [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. [32] A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, 301, Longman, Harlow, 1994. doi: 978-0582219779. [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. [21] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066. [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. [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. [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. [25] I. Podlubny, Fractional Differential Equations, Academic press, California, 1999.  doi: 10.1016/978-0-12-558840-9. [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. [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. [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. [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. [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. [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. [32] A. Viana, A local theory for a fractional reaction-diffusion equation, Commun. Contemp. Math, 21 (2019), 1850033, 26 pp. doi: 10.1142/S0219199718500335. [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. [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. [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.
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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