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  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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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