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doi: 10.3934/eect.2022029
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Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces

State Agency "Institute of applied mathematics and mechanics", Donetsk, Ukraine, Donetsk 83114, Ukraine

Received  September 2021 Revised  June 2022 Early access June 2022

We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another conditions we prove unique solvability of the Cauchy problem in the same spaces.

Citation: Sergey Degtyarev. Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. Evolution Equations and Control Theory, doi: 10.3934/eect.2022029
References:
[1]

R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal., 257 (2009), 2693-2722.  doi: 10.1016/j.jfa.2009.05.012.

[2]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Commun. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[3]

L. Brandolese and G. Karch, Far field asymptotics of solutions to convection equation with anomalous diffusion, J. Evol. Equ., 8 (2008), 307-326.  doi: 10.1007/s00028-008-0356-9.

[4]

S. Bu and G. Cai, Well-posedness of fractional degenerate differential equations with finite delay on vector-valued functional spaces, Math. Nachr., 291 (2018), 759-773.  doi: 10.1002/mana.201600502.

[5]

S. Bu and G. Cai, Well-posedness of degenerate fractional integro-differential equations in vector-valued functional spaces, Math. Nachr., 293 (2020), 1931-1946.  doi: 10.1002/mana.201900336.

[6]

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.

[7]

B. de Andrade, V. Van Au, D. O'Regan and N. H. Tuan, Well-posedness results for a class of semilinear time-fractional diffusion equations, Z. Angew. Math. Phys., 71 (2020), Paper No. 161, 24 pp. doi: 10.1007/s00033-020-01348-y.

[8]

S. P. Degtyarev, On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions, Evol. Equ. Control Theory, 4 (2015), 391-429.  doi: 10.3934/eect.2015.4.391.

[9]

S. P. Degtyarev, Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces, arXiv: 2109.11353.

[10]

M. G. Delgadino and S. Smith, Hölder estimates for fractional parabolic equations with critical divergence free drifts, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 577-604.  doi: 10.1016/j.anihpc.2017.06.004.

[11]

W. DengX. Wang and and P. Zhang, Anisotropic nonlocal diffusion operators for normal and anomalous dynamics, Multiscale Model. Sim., 18 (2020), 415-443.  doi: 10.1137/18M1184990.

[12]

H. Dong and H. Zhang, On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 40, 42 pp. doi: 10.1007/s00526-019-1482-7.

[13]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[14]

X. Ferna'ndez-Real and X. Ros-Oton, Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.  doi: 10.1016/j.jfa.2017.02.015.

[15]

C. G. Gal and M. Warma, Fractional-in-Time Semilinear Parabolic Equations and Applications, Mathématiques et Applications, Springer Nature Switzerland AG, 2020. doi: 10.1007/978-3-030-45043-4.

[16]

N. Garofalo, Fractional thoughts, in New Developments in the Analysis of Nonlocal Operators (eds. D. Danielli, A. Petrosyan and C. A. Pop), AMS Contemporary Mathematics, 723 (2019), 1–135. doi: 10.1090/conm/723/14569.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, Reprint of the 1998 Edition, Springer-Verlag, Berlin, Heidelberg, 2001.

[18]

K. K. Golovkin, On equivalent normalizations of fractional spaces, Amer. Math. Soc. Transl., 81 (1969), 257-280. 

[19]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics (eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien and New York, (1997), 223–276.

[20]

A. Greco and A. Iannizzotto, Existence and convexity of solutions of the fractional heat equation, Commun. Pure Appl. Anal., 16 (2017), 2201-2226.  doi: 10.3934/cpaa.2017109.

[21]

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.

[22]

K. Hisa, K. Ishige and J. Takahash, Existence of solutions for an inhomogeneous fractional semilinear heat equation, Nonlinear Anal., 199 (2020), 111920, 28 pp. doi: 10.1016/j.na.2020.111920.

[23]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Fractional Cauchy problems for infinite interval case, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3285-3304.  doi: 10.3934/dcdss.2020240.

[24]

J. Jia and K. Li, Maximum principles for a time–space fractional diffusion equation, Appl. Math. Lett., 62 (2016), 23-28.  doi: 10.1016/j.aml.2016.06.010.

[25]

E. KarimovM. Mamvhuev and M. Ruzhansky, Non-local initial problem for second order time-fractional and space-singular equation, Hokkaido Math. J., 49 (2020), 349-361.  doi: 10.14492/hokmj/1602036030.

[26]

J. Kemppainen, Positivity of the fundamental solution for fractional diffusion and wave equations, Math. Methods Appl. Sci., 44 (2021), 2468-2486.  doi: 10.1002/mma.5974.

[27]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 Amsterdam, Elsevier, 2006.

[28]

F. Kühn, Schauder estimates for equations associated with Lévy generators, Integr. Equ. Oper. Theory, 91 (2019), Paper No. 10, 21 pp. doi: 10.1007/s00020-019-2508-4.

[29]

O. A. Ladyzhenskaya, A Theorem on multiplicators in nonhomogeneous Hölder spaces and some of its applications, Journal of Mathematical Sciences, 115 (2003), 2792-2802.  doi: 10.1023/A:1023373920221.

[30]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Translations of Mathematical Monographs, 1968.

[31]

B. Li and X. Xie, Regularity of solutions to time fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3195-3210.  doi: 10.3934/dcdsb.2018340.

[32]

A. Lunardi and M. Röckner, Schauder theorems for a class of (pseudo-)differential operators on finite- and infinite-dimensional state spaces, J. London Math. Soc., 104 (2021), 492-540.  doi: 10.1112/jlms.12436.

[33]

L. Marino, Schauder estimates for degenerate Lévy Ornstein-Uhlenbeck operators, J. Math. Anal. Appl., 500 (2021), Paper No. 125168, 37 pp. doi: 10.1016/j.jmaa.2021.125168.

[34]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), no. 1, 77 pp. doi: 10.1016/S0370-1573(00)00070-3.

[35]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.

[36]

N. I. Muskhelishvili, Singular Integral Equations, Springer Science+Business Media B.V., Springer, Dordrecht, 1958.

[37]

A. de PabloF. Quirós and A. Rodríguez, Anisotropic nonlocal diffusion equations with singular forcing, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1167-1183.  doi: 10.1016/j.anihpc.2020.04.001.

[38]

J. Prüss, Evolutionary Integral Equations and Applications, Reprint of the 1993 Edition, Springer Basel Heidelberg New York Dordrecht London, 1993, Birkhauser, 2012. doi: 10.1007/978-3-0348-8570-6.

[39]

F. Punzo and E. Valdinoci, Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients, ESAIM Control Optim. Calc. Var., 24 (2018), 105-127.  doi: 10.1051/cocv/2016077.

[40]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York and London, 1993.

[41]

V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier– Stokes equations, Proc. Steklov Inst. Math., 70 (1964), 213-317. 

[42]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov., 83 (1965), 3-163. 

[43]

N. Tran BaoD. BaleanuD. Le Thi Minh and T. Nguyen Huy, Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel, Math. Meth. Appl. Sci., 43 (2020), 7208-7226.  doi: 10.1002/mma.6459.

[44]

H. Triebel, Theory of Function Spaces II, Modern Birkhäuser Classics, Birkhäuser, Basel, 2010.

[45]

S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Springer International Publishing, Switzerland, 2015. doi: 10.1007/978-3-319-20771-1.

[46]

K. Van Bockstal, Existence and uniqueness of a weak solution to a non-autonomous time-fractional diffusion equation (of distributed order), Appl. Math. Lett., 109 (2020), 106540, 8 pp. doi: 10.1016/j.aml.2020.106540.

[47]

K. Yabuta and M. Yang, Besov and Triebel–Lizorkin space estimates for fractional diffusion, Hiroshima Math. J., 48 (2018), 141-158.  doi: 10.32917/hmj/1533088828.

[48]

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.

show all references

References:
[1]

R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal., 257 (2009), 2693-2722.  doi: 10.1016/j.jfa.2009.05.012.

[2]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Commun. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[3]

L. Brandolese and G. Karch, Far field asymptotics of solutions to convection equation with anomalous diffusion, J. Evol. Equ., 8 (2008), 307-326.  doi: 10.1007/s00028-008-0356-9.

[4]

S. Bu and G. Cai, Well-posedness of fractional degenerate differential equations with finite delay on vector-valued functional spaces, Math. Nachr., 291 (2018), 759-773.  doi: 10.1002/mana.201600502.

[5]

S. Bu and G. Cai, Well-posedness of degenerate fractional integro-differential equations in vector-valued functional spaces, Math. Nachr., 293 (2020), 1931-1946.  doi: 10.1002/mana.201900336.

[6]

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.

[7]

B. de Andrade, V. Van Au, D. O'Regan and N. H. Tuan, Well-posedness results for a class of semilinear time-fractional diffusion equations, Z. Angew. Math. Phys., 71 (2020), Paper No. 161, 24 pp. doi: 10.1007/s00033-020-01348-y.

[8]

S. P. Degtyarev, On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions, Evol. Equ. Control Theory, 4 (2015), 391-429.  doi: 10.3934/eect.2015.4.391.

[9]

S. P. Degtyarev, Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces, arXiv: 2109.11353.

[10]

M. G. Delgadino and S. Smith, Hölder estimates for fractional parabolic equations with critical divergence free drifts, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 577-604.  doi: 10.1016/j.anihpc.2017.06.004.

[11]

W. DengX. Wang and and P. Zhang, Anisotropic nonlocal diffusion operators for normal and anomalous dynamics, Multiscale Model. Sim., 18 (2020), 415-443.  doi: 10.1137/18M1184990.

[12]

H. Dong and H. Zhang, On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 40, 42 pp. doi: 10.1007/s00526-019-1482-7.

[13]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[14]

X. Ferna'ndez-Real and X. Ros-Oton, Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.  doi: 10.1016/j.jfa.2017.02.015.

[15]

C. G. Gal and M. Warma, Fractional-in-Time Semilinear Parabolic Equations and Applications, Mathématiques et Applications, Springer Nature Switzerland AG, 2020. doi: 10.1007/978-3-030-45043-4.

[16]

N. Garofalo, Fractional thoughts, in New Developments in the Analysis of Nonlocal Operators (eds. D. Danielli, A. Petrosyan and C. A. Pop), AMS Contemporary Mathematics, 723 (2019), 1–135. doi: 10.1090/conm/723/14569.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of the Second Order, Reprint of the 1998 Edition, Springer-Verlag, Berlin, Heidelberg, 2001.

[18]

K. K. Golovkin, On equivalent normalizations of fractional spaces, Amer. Math. Soc. Transl., 81 (1969), 257-280. 

[19]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics (eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien and New York, (1997), 223–276.

[20]

A. Greco and A. Iannizzotto, Existence and convexity of solutions of the fractional heat equation, Commun. Pure Appl. Anal., 16 (2017), 2201-2226.  doi: 10.3934/cpaa.2017109.

[21]

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.

[22]

K. Hisa, K. Ishige and J. Takahash, Existence of solutions for an inhomogeneous fractional semilinear heat equation, Nonlinear Anal., 199 (2020), 111920, 28 pp. doi: 10.1016/j.na.2020.111920.

[23]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Fractional Cauchy problems for infinite interval case, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3285-3304.  doi: 10.3934/dcdss.2020240.

[24]

J. Jia and K. Li, Maximum principles for a time–space fractional diffusion equation, Appl. Math. Lett., 62 (2016), 23-28.  doi: 10.1016/j.aml.2016.06.010.

[25]

E. KarimovM. Mamvhuev and M. Ruzhansky, Non-local initial problem for second order time-fractional and space-singular equation, Hokkaido Math. J., 49 (2020), 349-361.  doi: 10.14492/hokmj/1602036030.

[26]

J. Kemppainen, Positivity of the fundamental solution for fractional diffusion and wave equations, Math. Methods Appl. Sci., 44 (2021), 2468-2486.  doi: 10.1002/mma.5974.

[27]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 Amsterdam, Elsevier, 2006.

[28]

F. Kühn, Schauder estimates for equations associated with Lévy generators, Integr. Equ. Oper. Theory, 91 (2019), Paper No. 10, 21 pp. doi: 10.1007/s00020-019-2508-4.

[29]

O. A. Ladyzhenskaya, A Theorem on multiplicators in nonhomogeneous Hölder spaces and some of its applications, Journal of Mathematical Sciences, 115 (2003), 2792-2802.  doi: 10.1023/A:1023373920221.

[30]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Translations of Mathematical Monographs, 1968.

[31]

B. Li and X. Xie, Regularity of solutions to time fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3195-3210.  doi: 10.3934/dcdsb.2018340.

[32]

A. Lunardi and M. Röckner, Schauder theorems for a class of (pseudo-)differential operators on finite- and infinite-dimensional state spaces, J. London Math. Soc., 104 (2021), 492-540.  doi: 10.1112/jlms.12436.

[33]

L. Marino, Schauder estimates for degenerate Lévy Ornstein-Uhlenbeck operators, J. Math. Anal. Appl., 500 (2021), Paper No. 125168, 37 pp. doi: 10.1016/j.jmaa.2021.125168.

[34]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), no. 1, 77 pp. doi: 10.1016/S0370-1573(00)00070-3.

[35]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.

[36]

N. I. Muskhelishvili, Singular Integral Equations, Springer Science+Business Media B.V., Springer, Dordrecht, 1958.

[37]

A. de PabloF. Quirós and A. Rodríguez, Anisotropic nonlocal diffusion equations with singular forcing, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1167-1183.  doi: 10.1016/j.anihpc.2020.04.001.

[38]

J. Prüss, Evolutionary Integral Equations and Applications, Reprint of the 1993 Edition, Springer Basel Heidelberg New York Dordrecht London, 1993, Birkhauser, 2012. doi: 10.1007/978-3-0348-8570-6.

[39]

F. Punzo and E. Valdinoci, Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients, ESAIM Control Optim. Calc. Var., 24 (2018), 105-127.  doi: 10.1051/cocv/2016077.

[40]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York and London, 1993.

[41]

V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier– Stokes equations, Proc. Steklov Inst. Math., 70 (1964), 213-317. 

[42]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov., 83 (1965), 3-163. 

[43]

N. Tran BaoD. BaleanuD. Le Thi Minh and T. Nguyen Huy, Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel, Math. Meth. Appl. Sci., 43 (2020), 7208-7226.  doi: 10.1002/mma.6459.

[44]

H. Triebel, Theory of Function Spaces II, Modern Birkhäuser Classics, Birkhäuser, Basel, 2010.

[45]

S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Springer International Publishing, Switzerland, 2015. doi: 10.1007/978-3-319-20771-1.

[46]

K. Van Bockstal, Existence and uniqueness of a weak solution to a non-autonomous time-fractional diffusion equation (of distributed order), Appl. Math. Lett., 109 (2020), 106540, 8 pp. doi: 10.1016/j.aml.2020.106540.

[47]

K. Yabuta and M. Yang, Besov and Triebel–Lizorkin space estimates for fractional diffusion, Hiroshima Math. J., 48 (2018), 141-158.  doi: 10.32917/hmj/1533088828.

[48]

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.

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