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doi: 10.3934/eect.2021050
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Linear subdiffusion in weighted fractional Hölder spaces

1. 

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, G.Batyuka str. 19, 84100 Sloviansk, Ukraine

2. 

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, G.Batyuka str. 19, 84100 Sloviansk, Ukraine and, Institute of Hydromechanics of NAS of Ukraine, Zhelyabova str. 8/4, 03057 Kyiv, Ukraine

* Corresponding author: N. Vasylyeva

Received  January 2021 Revised  May 2021 Early access September 2021

For
$ \nu\in(0,1) $
, we investigate the nonautonomous subdiffusion equation:
$ \mathbf{D}_{t}^{\nu}u-\mathcal{L}u = f(x,t), $
where
$ \mathbf{D}_{t}^{\nu} $
is the Caputofractional derivative and
$ \mathcal{L} $
is a uniformly ellipticoperator with smooth coefficients depending on time. Undersuitable conditions on the given data and a minimal number (i.e.the necessary number) of compatibility conditions, the globalclassical solvability to the related initial-boundary valueproblems are established in the weighted fractional Hölderspaces.
Citation: Mykola Krasnoschok, Nataliya Vasylyeva. Linear subdiffusion in weighted fractional Hölder spaces. Evolution Equations & Control Theory, doi: 10.3934/eect.2021050
References:
[1]

M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55 U. S. Government Printing Office, Washington, D. C., 1964  Google Scholar

[2]

R. L. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelastisity, J. Rheol., 27 (1983), 201-210.   Google Scholar

[3]

E. BazhlekovaB. JinR. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.  Google Scholar

[4]

V. S. Belonosov, Estimates of solutions of parabolic systems in weighted Hölder classes and some of their applications, Mat. SSSR Sb., 38 (1981), 151-173.   Google Scholar

[5]

G. I. Bizhanova, Solution in a weighted Hölder space of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the conjugation condition, Algebra i Analiz, 6 (1994), 64-94.   Google Scholar

[6]

G. I. Bizhanova and V. A. Solonnikov, On the solvability of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the boundary condition in a weighted Hölder space of functions, Algebra i Analiz, 5 (1993), 109-142.   Google Scholar

[7]

M. Caputo, Models of flux in porous media with memory, Water Resour. Res., 36 (2000), 693-705.  doi: 10.1029/1999WR900299.  Google Scholar

[8]

A. Carbotti, S. Dipierro and E. Valdinoci, Local Density of Solutions to Fractional Equations, De Gruyter Studies in Mathematics, 2019. doi: 10.1515/9783110664355.  Google Scholar

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P. ClémentG. Gripenberg and S-O. Londen, Schauder estimates for equations with fractional derivative, Trans. Amer. Math. Soc., 352 (2000), 2239-2260.  doi: 10.1090/S0002-9947-00-02507-1.  Google Scholar

[10]

P. ClémentS-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations, 196 (2004), 418-447.  doi: 10.1016/j.jde.2003.07.014.  Google Scholar

[11]

R. M. Dzhafarov and N. V. Krasnoschok, The Cauchy problem for the fractional diffusion equation in a weighted Hölder space, Siberian Math. J., 59 (2018), 1034-1051.  doi: 10.1134/S0037446618060071.  Google Scholar

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N. Engheia, On the role of fractional calculus in electromagnetic theory, IEEE Antennas and Propagation Mag., 39 (1997), 35-46.  doi: 10.1109/74.632994.  Google Scholar

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D. Guidetti, On maximal regularity for the Cauchy-Dirichlet parabolic problem with fractional time derivative, J. Math. Anal. Appl., 476 (2019), 637-664.  doi: 10.1016/j.jmaa.2019.04.004.  Google Scholar

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J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electronic J. Diff. Equat., 2016 (2016), 1-28.   Google Scholar

[15]

J. Janno and K. Kasemets, Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation, Inverse Probl. Imaging, 11 (2017), 125-149.  doi: 10.3934/ipi.2017007.  Google Scholar

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J. Kemppainen and K. Ruotsalainen, Boundary integral solution of the time-fractional diffusion equation, Integr. Equ. Oper. Theory, 64 (2009), 239-249.  doi: 10.1007/s00020-009-1687-9.  Google Scholar

[17]

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

[18]

I. KimK-H. Kim and S. Lim, An $L_{q}(L_{p})-$ theory for the time fractional evolution equations with variable coefficients, Advances Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.  Google Scholar

[19]

N. Kinash and J. Janno, Inverse problems for a generalized subdiffusion equation with final overdetermination, Math. Model. Anal., 24 (2019), 236-262.  doi: 10.3846/mma.2019.016.  Google Scholar

[20]

A. N. Kochubei, The Cauchy problem for evolution equations of fractional order, Differential Equations, 25 (1989), 967-974.   Google Scholar

[21]

A. N. Kochubei, Diffusion of fractional order, Differential Equations, 26 (1990), 485-492.   Google Scholar

[22]

M. Krasnoschok, Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion, J. Math. Phys. Anal. Geometry, 12 (2016), 48-77.  doi: 10.15407/mag12.01.048.  Google Scholar

[23]

M. KrasnoschokV. Pata and N. Vasylyeva, Solvability of linear boundary value problems for subdiffusion equation with memory, J. Integral Equations Appl., 30 (2018), 417-445.  doi: 10.1216/JIE-2018-30-3-417.  Google Scholar

[24]

M. KrasnoschokV. Pata and N. Vasylyeva, Semilinear subdiffusion with memory in multidimensional domains, Math. Nachr., 292 (2019), 1490-1513.  doi: 10.1002/mana.201700405.  Google Scholar

[25]

M. KrasnoschokV. PataS. V. Siryk and N. Vasylyeva, Equivalent definitions of Caputo derivatives and applications to subdiffusion equations, Dyn. Partial Differ. Equ., 17 (2020), 383-402.  doi: 10.4310/DPDE.2020.v17.n4.a4.  Google Scholar

[26]

M. Krasnoschok, S. Pereverzyev, S. V. Siryk and N. Vasylyeva, Regularized reconstruction of the order in semilinear subdiffusion with memory, Inverse Problems and Related Topics, Springer Proceedings in Mathematics & Statistics, 310 (2020), 205–236. doi: 10.1007/978-981-15-1592-7\_10.  Google Scholar

[27]

M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlinear Stud., 20 (2013), 591-621.   Google Scholar

[28] O. A. LadyzhenskaiaV. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations, Academic Press, New York, 1968.   Google Scholar
[29]

Z. LiX. Huang and M. Yamamoto, A stability result for the determination of order in time-fractional diffusion equations, J. Inverse Ill-Posed Probl., 28 (2019), 379-388.  doi: 10.1515/jiip-2018-0079.  Google Scholar

[30]

C. Lizama and G. M. Guérékata, Bounded mild solutions for semilinear integro-differential equations, J. Integral. Equations Appl., 5 (1993), 75-78.   Google Scholar

[31]

Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal., 19 (2016), 676-695.  doi: 10.1515/fca-2016-0036.  Google Scholar

[32]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications 16, Basel: Birkhäuser Verlag, 1995.  Google Scholar

[33]

R. L. Magin, Fractional Calculus in Bioengineering, Redding, Begell House, 2006. Google Scholar

[34]

W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617.  Google Scholar

[35]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43. Walter de Gruyter & Co., Berlin, 2012.  Google Scholar

[36]

C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York-Berlin 1970.  Google Scholar

[37]

G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Appl. Anal., 91 (2012), 15-34.  doi: 10.1080/00036811.2010.534730.  Google Scholar

[38]

J. MuB. Ahmad and S. Hueng, Existence and regularity of solutions to time-fractional diffusion equations, Comput. Math. Appl., 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.  Google Scholar

[39]

J. NakagawaK. Sakamoto and M. Yamamoto, Overview to mathematical analysis for fractional diffusion equations – new mathematical aspects motivated by industrial collaboration, J. Math-for-Ind., 2 (2010), 99-108.   Google Scholar

[40]

R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity, J. Differential Equations, 255 (2013), 3284-3304.  doi: 10.1016/j.jde.2013.07.035.  Google Scholar

[41]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 2012. Google Scholar

[42]

A. V. Pskhu, Partial Differential Equations of the Fractional Order, Nauka, Moscow, 2005.  Google Scholar

[43]

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

[44]

V. A. Solonnikov, Estimates for the solution of the second initial-boundary value problem for the Stokes systems in spaces of functions with Hölder-continous derivatives with respect to the space variables, J. Math. Sci., 109 (2002), 1997-2017. doi: 10.1023/A:1014456711451.  Google Scholar

[45]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 543-577.  doi: 10.1007/s00030-014-0295-9.  Google Scholar

[46]

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

[47]

R. Zacher, Maximal regularity of type $L_{p}$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

show all references

References:
[1]

M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55 U. S. Government Printing Office, Washington, D. C., 1964  Google Scholar

[2]

R. L. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelastisity, J. Rheol., 27 (1983), 201-210.   Google Scholar

[3]

E. BazhlekovaB. JinR. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math., 131 (2015), 1-31.  doi: 10.1007/s00211-014-0685-2.  Google Scholar

[4]

V. S. Belonosov, Estimates of solutions of parabolic systems in weighted Hölder classes and some of their applications, Mat. SSSR Sb., 38 (1981), 151-173.   Google Scholar

[5]

G. I. Bizhanova, Solution in a weighted Hölder space of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the conjugation condition, Algebra i Analiz, 6 (1994), 64-94.   Google Scholar

[6]

G. I. Bizhanova and V. A. Solonnikov, On the solvability of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the boundary condition in a weighted Hölder space of functions, Algebra i Analiz, 5 (1993), 109-142.   Google Scholar

[7]

M. Caputo, Models of flux in porous media with memory, Water Resour. Res., 36 (2000), 693-705.  doi: 10.1029/1999WR900299.  Google Scholar

[8]

A. Carbotti, S. Dipierro and E. Valdinoci, Local Density of Solutions to Fractional Equations, De Gruyter Studies in Mathematics, 2019. doi: 10.1515/9783110664355.  Google Scholar

[9]

P. ClémentG. Gripenberg and S-O. Londen, Schauder estimates for equations with fractional derivative, Trans. Amer. Math. Soc., 352 (2000), 2239-2260.  doi: 10.1090/S0002-9947-00-02507-1.  Google Scholar

[10]

P. ClémentS-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations, 196 (2004), 418-447.  doi: 10.1016/j.jde.2003.07.014.  Google Scholar

[11]

R. M. Dzhafarov and N. V. Krasnoschok, The Cauchy problem for the fractional diffusion equation in a weighted Hölder space, Siberian Math. J., 59 (2018), 1034-1051.  doi: 10.1134/S0037446618060071.  Google Scholar

[12]

N. Engheia, On the role of fractional calculus in electromagnetic theory, IEEE Antennas and Propagation Mag., 39 (1997), 35-46.  doi: 10.1109/74.632994.  Google Scholar

[13]

D. Guidetti, On maximal regularity for the Cauchy-Dirichlet parabolic problem with fractional time derivative, J. Math. Anal. Appl., 476 (2019), 637-664.  doi: 10.1016/j.jmaa.2019.04.004.  Google Scholar

[14]

J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electronic J. Diff. Equat., 2016 (2016), 1-28.   Google Scholar

[15]

J. Janno and K. Kasemets, Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation, Inverse Probl. Imaging, 11 (2017), 125-149.  doi: 10.3934/ipi.2017007.  Google Scholar

[16]

J. Kemppainen and K. Ruotsalainen, Boundary integral solution of the time-fractional diffusion equation, Integr. Equ. Oper. Theory, 64 (2009), 239-249.  doi: 10.1007/s00020-009-1687-9.  Google Scholar

[17]

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

[18]

I. KimK-H. Kim and S. Lim, An $L_{q}(L_{p})-$ theory for the time fractional evolution equations with variable coefficients, Advances Math., 306 (2017), 123-176.  doi: 10.1016/j.aim.2016.08.046.  Google Scholar

[19]

N. Kinash and J. Janno, Inverse problems for a generalized subdiffusion equation with final overdetermination, Math. Model. Anal., 24 (2019), 236-262.  doi: 10.3846/mma.2019.016.  Google Scholar

[20]

A. N. Kochubei, The Cauchy problem for evolution equations of fractional order, Differential Equations, 25 (1989), 967-974.   Google Scholar

[21]

A. N. Kochubei, Diffusion of fractional order, Differential Equations, 26 (1990), 485-492.   Google Scholar

[22]

M. Krasnoschok, Solvability in Hölder space of an initial boundary value problem for the time-fractional diffusion, J. Math. Phys. Anal. Geometry, 12 (2016), 48-77.  doi: 10.15407/mag12.01.048.  Google Scholar

[23]

M. KrasnoschokV. Pata and N. Vasylyeva, Solvability of linear boundary value problems for subdiffusion equation with memory, J. Integral Equations Appl., 30 (2018), 417-445.  doi: 10.1216/JIE-2018-30-3-417.  Google Scholar

[24]

M. KrasnoschokV. Pata and N. Vasylyeva, Semilinear subdiffusion with memory in multidimensional domains, Math. Nachr., 292 (2019), 1490-1513.  doi: 10.1002/mana.201700405.  Google Scholar

[25]

M. KrasnoschokV. PataS. V. Siryk and N. Vasylyeva, Equivalent definitions of Caputo derivatives and applications to subdiffusion equations, Dyn. Partial Differ. Equ., 17 (2020), 383-402.  doi: 10.4310/DPDE.2020.v17.n4.a4.  Google Scholar

[26]

M. Krasnoschok, S. Pereverzyev, S. V. Siryk and N. Vasylyeva, Regularized reconstruction of the order in semilinear subdiffusion with memory, Inverse Problems and Related Topics, Springer Proceedings in Mathematics & Statistics, 310 (2020), 205–236. doi: 10.1007/978-981-15-1592-7\_10.  Google Scholar

[27]

M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlinear Stud., 20 (2013), 591-621.   Google Scholar

[28] O. A. LadyzhenskaiaV. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Parabolic Equations, Academic Press, New York, 1968.   Google Scholar
[29]

Z. LiX. Huang and M. Yamamoto, A stability result for the determination of order in time-fractional diffusion equations, J. Inverse Ill-Posed Probl., 28 (2019), 379-388.  doi: 10.1515/jiip-2018-0079.  Google Scholar

[30]

C. Lizama and G. M. Guérékata, Bounded mild solutions for semilinear integro-differential equations, J. Integral. Equations Appl., 5 (1993), 75-78.   Google Scholar

[31]

Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal., 19 (2016), 676-695.  doi: 10.1515/fca-2016-0036.  Google Scholar

[32]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications 16, Basel: Birkhäuser Verlag, 1995.  Google Scholar

[33]

R. L. Magin, Fractional Calculus in Bioengineering, Redding, Begell House, 2006. Google Scholar

[34]

W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138.  doi: 10.1017/S1446181111000617.  Google Scholar

[35]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43. Walter de Gruyter & Co., Berlin, 2012.  Google Scholar

[36]

C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York-Berlin 1970.  Google Scholar

[37]

G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Appl. Anal., 91 (2012), 15-34.  doi: 10.1080/00036811.2010.534730.  Google Scholar

[38]

J. MuB. Ahmad and S. Hueng, Existence and regularity of solutions to time-fractional diffusion equations, Comput. Math. Appl., 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.  Google Scholar

[39]

J. NakagawaK. Sakamoto and M. Yamamoto, Overview to mathematical analysis for fractional diffusion equations – new mathematical aspects motivated by industrial collaboration, J. Math-for-Ind., 2 (2010), 99-108.   Google Scholar

[40]

R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity, J. Differential Equations, 255 (2013), 3284-3304.  doi: 10.1016/j.jde.2013.07.035.  Google Scholar

[41]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 2012. Google Scholar

[42]

A. V. Pskhu, Partial Differential Equations of the Fractional Order, Nauka, Moscow, 2005.  Google Scholar

[43]

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

[44]

V. A. Solonnikov, Estimates for the solution of the second initial-boundary value problem for the Stokes systems in spaces of functions with Hölder-continous derivatives with respect to the space variables, J. Math. Sci., 109 (2002), 1997-2017. doi: 10.1023/A:1014456711451.  Google Scholar

[45]

N. Vasylyeva and L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 543-577.  doi: 10.1007/s00030-014-0295-9.  Google Scholar

[46]

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

[47]

R. Zacher, Maximal regularity of type $L_{p}$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

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