September  2020, 9(3): 891-914. doi: 10.3934/eect.2020038

Nonlocal final value problem governed by semilinear anomalous diffusion equations

1. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

2. 

Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

* Corresponding author: Dinh-Ke TRAN (ketd@hnue.edu.vn)

Received  June 2019 Revised  November 2019 Published  September 2020 Early access  March 2020

Our goal is to establish some sufficient conditions for the solvability of the nonlocal final value problem involving a class of partial differential equations, which describes the anomalous diffusion phenomenon. Our analysis is based on the theory of completely positive functions, resolvent operators and fixed point arguments in suitable function spaces. Especially, utilizing the regularity of resolvent operators, we are able to deal with non-Lipschitz cases. The obtained results, in particular, extend recent ones proved for fractional diffusion equations.

Citation: Dinh-Ke Tran, Tran-Phuong-Thuy Lam. Nonlocal final value problem governed by semilinear anomalous diffusion equations. Evolution Equations and Control Theory, 2020, 9 (3) : 891-914. doi: 10.3934/eect.2020038
References:
[1]

Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

[2]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[3]

J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $mathbb{R}^{d}$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.

[4]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.

[5] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.
[6]

R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl., 22 (1968), 319-340.  doi: 10.1016/0022-247X(68)90176-5.

[7]

S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. Math. Sci., 57 (2003), 3609-3632.  doi: 10.1155/S0161171203211455.

[8]

J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666.  doi: 10.3934/dcds.2019026.

[9]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[10]

N. H. TuanM. KiraneB. Bin-Mohsin and P. T. M. Tam, Filter regularization for final value fractional diffusion problem with deterministic and random noise, Comput. Math. Appl., 74 (2017), 1340-1361.  doi: 10.1016/j.camwa.2017.06.014.

[11]

N. H. TuanL. D. LongV. T. Nguyen and T. Tran, On a final value problem for the time-fractional diffusion equation with inhomogeneous source, Inverse Probl. Sci. Eng., 25 (2017), 1367-1395.  doi: 10.1080/17415977.2016.1259316.

[12]

N. H. TuanT. B. NgocL. N. Huynh and M. Kirane, Existence and uniqueness of mild solution of time-fractional semilinear differential equations with a nonlocal final condition, Comput. Math. Appl., 78 (2019), 1651-1668.  doi: 10.1016/j.camwa.2018.11.007.

[13]

N. H. TuanL. N. HuynhT. B. Ngoc and Y. Zhou, On a backward problem for nonlinear fractional diffusion equations, Appl. Math. Lett., 92 (2019), 76-84.  doi: 10.1016/j.aml.2018.11.015.

[14]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[15]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

[16]

I. I. Vrabie, $C_0$-Semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003.

[17]

F. YangY.-P. Ren and X.-X. Li, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source, Math. Methods Appl. Sci., 41 (2018), 1774-1795.  doi: 10.1002/mma.4705.

[18]

M. Yang and J. J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math., 66 (2013), 45-58.  doi: 10.1016/j.apnum.2012.11.009.

[19]

H. W. Zhang and X. J. Zhang, Generalized Tikhonov method for the final value problem of time-fractional diffusion equation, Int. J. Comput. Math., 94 (2017), 66-78.  doi: 10.1080/00207160.2015.1089354.

show all references

References:
[1]

Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

[2]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[3]

J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $mathbb{R}^{d}$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.

[4]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.

[5] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.
[6]

R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl., 22 (1968), 319-340.  doi: 10.1016/0022-247X(68)90176-5.

[7]

S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. Math. Sci., 57 (2003), 3609-3632.  doi: 10.1155/S0161171203211455.

[8]

J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666.  doi: 10.3934/dcds.2019026.

[9]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[10]

N. H. TuanM. KiraneB. Bin-Mohsin and P. T. M. Tam, Filter regularization for final value fractional diffusion problem with deterministic and random noise, Comput. Math. Appl., 74 (2017), 1340-1361.  doi: 10.1016/j.camwa.2017.06.014.

[11]

N. H. TuanL. D. LongV. T. Nguyen and T. Tran, On a final value problem for the time-fractional diffusion equation with inhomogeneous source, Inverse Probl. Sci. Eng., 25 (2017), 1367-1395.  doi: 10.1080/17415977.2016.1259316.

[12]

N. H. TuanT. B. NgocL. N. Huynh and M. Kirane, Existence and uniqueness of mild solution of time-fractional semilinear differential equations with a nonlocal final condition, Comput. Math. Appl., 78 (2019), 1651-1668.  doi: 10.1016/j.camwa.2018.11.007.

[13]

N. H. TuanL. N. HuynhT. B. Ngoc and Y. Zhou, On a backward problem for nonlinear fractional diffusion equations, Appl. Math. Lett., 92 (2019), 76-84.  doi: 10.1016/j.aml.2018.11.015.

[14]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[15]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

[16]

I. I. Vrabie, $C_0$-Semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003.

[17]

F. YangY.-P. Ren and X.-X. Li, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source, Math. Methods Appl. Sci., 41 (2018), 1774-1795.  doi: 10.1002/mma.4705.

[18]

M. Yang and J. J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math., 66 (2013), 45-58.  doi: 10.1016/j.apnum.2012.11.009.

[19]

H. W. Zhang and X. J. Zhang, Generalized Tikhonov method for the final value problem of time-fractional diffusion equation, Int. J. Comput. Math., 94 (2017), 66-78.  doi: 10.1080/00207160.2015.1089354.

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