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Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps
Time-fractional equations with reaction terms: Fundamental solutions and asymptotics
1. | Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia |
2. | Dipartimento di Matematica e Fisica, Università della Campania "Luigi Vanvitelli", Viale Lincoln 5, 81100 Caserta, Italy |
3. | Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia |
4. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy |
We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space.
We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, $ \frac12 $. In this situation, we prove that the speed of invasion of the fundamental solution is at least "almost of square root type", namely it is larger than $ ct^\beta $ for any given $ c>0 $ and $ \beta\in\left(0,\frac12\right) $.
References:
[1] |
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in Contemporary Research in Elliptic PDEs and Related Topics, vol. 33 of Springer INdAM Ser., Springer, Cham, 2019, 1–105. https://link.springer.com/chapter/10.1007/978-3-030-18921-1_1. |
[2] |
E. Affili and E. Valdinoci,
Decay estimates for evolution equations with classical and fractional time-derivatives, J. Differential Equations, 266 (2019), 4027-4060.
doi: 10.1016/j.jde.2018.09.031. |
[3] |
V. E. Arkhincheev and E. M. Baskin, Anomalous diffusion and drift in a comb model of percolation clusters, J. Exp.Theor. Phys., 73 (1991), 161–165. http://www.jetp.ac.ru/cgi-bin/e/index/e/73/1/p161?a=list. |
[4] |
X. Cabré, A.-C. Coulon and J.-M. Roquejoffre,
Propagation in Fisher-KPP type equations with fractional diffusion in periodic media, C. R. Math. Acad. Sci. Paris, 350 (2012), 885-890.
doi: 10.1016/j.crma.2012.10.007. |
[5] |
X. Cabré and J.-M. Roquejoffre,
The influence of fractional diffusion in {F}isher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.
doi: 10.1007/s00220-013-1682-5. |
[6] |
M. Caputo, Linear models of dissipation whose {$Q$} is almost frequency independent. II, Fract. Calc. Appl. Anal., 11 (2008), 4–14, Reprinted from Geophys. J. R. Astr. Soc., 13 (1967), 529–539. https://www.annalsofgeophysics.eu/index.php/annals/article/viewFile/5051/5122. |
[7] |
A. Carbotti, S. Dipierro and E. Valdinoci, Local density of solutions to fractional equations, De Gruyter Studies in Mathematics 74. De Gruyter, Berlin, https://www.degruyter.com/view/product/534026. |
[8] |
W. E. Deming and C. G. Colcord, The minimum in the gamma function, Nature, 135 (1935), 917.
doi: 10.1038/135917b0. |
[9] |
K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type. Vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14574-2. |
[10] |
S. Dipierro and E. Valdinoci,
A simple mathematical model inspired by the {P}urkinje cells: from delayed travelling waves to fractional diffusion, Bull. Math. Biol., 80 (2018), 1849-1870.
doi: 10.1007/s11538-018-0437-z. |
[11] |
S. Dipierro, E. Valdinoci and V. Vespri,
Decay estimates for evolutionary equations with fractional time-diffusion, J. Evol. Equ., 19 (2019), 435-462.
doi: 10.1007/s00028-019-00482-z. |
[12] |
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. |
[13] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. {V}ol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955, https://mathscinet.ams.org/mathscinet-getitem?mr=0066496, Based, in part, on notes left by Harry Bateman. |
[14] |
F. Ferrari, {W}eyl and {M}archaud {D}erivatives: A forgotten history, Mathematics, 6 (2018), 6. https://www.mdpi.com/2227-7390/6/1/6.
doi: 10.3390/math6010006. |
[15] |
I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967, Theory of differential equations, Translated from the Russian by Meinhard E. Mayer. https://www.ams.org/books/chel/379/chel379-endmatter.pdf. |
[16] |
C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates,
The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.
doi: 10.1016/j.cnsns.2017.04.001. |
[17] |
S. L. Kalla and B. Ross, The development of functional relations by means of fractional operators, in Fractional Calculus ({G}lasgow, 1984), vol. 138 of Res. Notes in Math., Pitman, Boston, MA, 1985, 32–43. |
[18] |
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher,
Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.
doi: 10.1007/s00208-015-1356-z. |
[19] |
J. Kemppainen, J. Siljander and R. Zacher,
Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.
doi: 10.1016/j.jde.2017.02.030. |
[20] |
J. Kemppainen and R. Zacher,
Long-time behavior of non-local in time {F}okker–{P}lanck equations via the entropy method, Math. Models Methods Appl. Sci., 29 (2019), 209-235.
doi: 10.1142/S0218202519500076. |
[21] |
Y. Luchko,
Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.
|
[22] |
Y. Luchko,
Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.
doi: 10.1016/j.jmaa.2010.08.048. |
[23] |
F. Mainardi,
On some properties of the Mittag-Leffler function {$E_\alpha(-t^\alpha)$}, completely monotone for {$t>0$} with $0 < \alpha < 1$, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2267-2278.
doi: 10.3934/dcdsb.2014.19.2267. |
[24] |
F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153–192. https://arXiv.org/pdf/cond-mat/0702419.pdf. |
[25] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, vol. 43 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2012. https://www.degruyter.com/view/product/129781. |
[26] |
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999.
![]() |
[27] |
J.-M. Roquejoffre and A. Tarfulea,
Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion, J. Math. Pures Appl. (9), 108 (2017), 399-424.
doi: 10.1016/j.matpur.2017.07.001. |
[28] |
B. Ross, The Development, Theory and Application of the Gamma-function and a Profile of Fractional-calculus, ProQuest LLC, Ann Arbor, MI, 1974, Thesis (Ph.D.)–New York University. http://gateway.proquest.com.pros.lib.unimi.it/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7417154 |
[29] |
B. Ross,
The development of fractional calculus 1695–1900, Historia Math., 4 (1977), 75-89.
doi: 10.1016/0315-0860(77)90039-8. |
[30] |
B. Ross,
Origins of fractional calculus and some applications, Internat. J. Math. Statist. Sci., 1 (1992), 21-34.
|
[31] |
J. Sánchez and V. Vergara,
Long-time behavior of bounded global solutions to systems of nonlinear integro-differential equations, Asymptot. Anal., 85 (2013), 167-178.
doi: 10.3233/ASY-131180. |
[32] |
J. Sánchez and V. Vergara,
Long-time behavior of nonlinear integro-differential evolution equations, Nonlinear Anal., 91 (2013), 20-31.
doi: 10.1016/j.na.2013.06.006. |
[33] |
T. Sandev, A. Schulz, H. Kantz and A. Iomin,
Heterogeneous diffusion in comb and fractal grid structures, Chaos Solitons Fractals, 114 (2018), 551-555.
doi: 10.1016/j.chaos.2017.04.041. |
[34] |
F. Santamaria, S. Wils, E. D. Schutter and G. J. Augustine, The diffusional properties of dendrites depend on the density of dendritic spines, Eur. J. Neurosci., 34 (2011), 561–568. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3156966/.
doi: 10.1111/j.1460-9568.2011.07785.x. |
[35] |
H. Schiessel, C. Friedrich and A. Blumen, Applications to problems in polymer physics and rheology, in Applications of Fractional Calculus in Physics, World Sci. Publ., River Edge, NJ, 2000,331–376.
doi: 10.1142/9789812817747_0007. |
[36] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with {C}aputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
[37] |
V. Vergara and R. Zacher,
A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73 (2010), 3572-3585.
doi: 10.1016/j.na.2010.07.039. |
[38] |
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. |
[39] |
R. Wong and Y.-Q. Zhao,
Exponential asymptotics of the {M}ittag-{L}effler function, Constr. Approx., 18 (2002), 355-385.
doi: 10.1007/s00365-001-0019-3. |
[40] |
R. Zacher,
Maximal regularity of type {$L_p$} for abstract parabolic {V}olterra equations, J. Evol. Equ., 5 (2005), 79-103.
doi: 10.1007/s00028-004-0161-z. |
[41] |
R. Zacher,
Weak solutions of abstract evolutionary integro-differential equations in {H}ilbert spaces, Funkcial. Ekvac., 52 (2009), 1-18.
doi: 10.1619/fesi.52.1. |
[42] |
R. Zacher, Time fractional diffusion equations: Solution concepts, regularity, and long-time behavior, in Handbook of Fractional Calculus with Applications. {V}ol. 2, De Gruyter, Berlin, 2019,159–179. https://www.degruyter.com/viewbooktoc/product/497030.
doi: 10.1515/9783110571660-008. |
show all references
References:
[1] |
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in Contemporary Research in Elliptic PDEs and Related Topics, vol. 33 of Springer INdAM Ser., Springer, Cham, 2019, 1–105. https://link.springer.com/chapter/10.1007/978-3-030-18921-1_1. |
[2] |
E. Affili and E. Valdinoci,
Decay estimates for evolution equations with classical and fractional time-derivatives, J. Differential Equations, 266 (2019), 4027-4060.
doi: 10.1016/j.jde.2018.09.031. |
[3] |
V. E. Arkhincheev and E. M. Baskin, Anomalous diffusion and drift in a comb model of percolation clusters, J. Exp.Theor. Phys., 73 (1991), 161–165. http://www.jetp.ac.ru/cgi-bin/e/index/e/73/1/p161?a=list. |
[4] |
X. Cabré, A.-C. Coulon and J.-M. Roquejoffre,
Propagation in Fisher-KPP type equations with fractional diffusion in periodic media, C. R. Math. Acad. Sci. Paris, 350 (2012), 885-890.
doi: 10.1016/j.crma.2012.10.007. |
[5] |
X. Cabré and J.-M. Roquejoffre,
The influence of fractional diffusion in {F}isher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.
doi: 10.1007/s00220-013-1682-5. |
[6] |
M. Caputo, Linear models of dissipation whose {$Q$} is almost frequency independent. II, Fract. Calc. Appl. Anal., 11 (2008), 4–14, Reprinted from Geophys. J. R. Astr. Soc., 13 (1967), 529–539. https://www.annalsofgeophysics.eu/index.php/annals/article/viewFile/5051/5122. |
[7] |
A. Carbotti, S. Dipierro and E. Valdinoci, Local density of solutions to fractional equations, De Gruyter Studies in Mathematics 74. De Gruyter, Berlin, https://www.degruyter.com/view/product/534026. |
[8] |
W. E. Deming and C. G. Colcord, The minimum in the gamma function, Nature, 135 (1935), 917.
doi: 10.1038/135917b0. |
[9] |
K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type. Vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-14574-2. |
[10] |
S. Dipierro and E. Valdinoci,
A simple mathematical model inspired by the {P}urkinje cells: from delayed travelling waves to fractional diffusion, Bull. Math. Biol., 80 (2018), 1849-1870.
doi: 10.1007/s11538-018-0437-z. |
[11] |
S. Dipierro, E. Valdinoci and V. Vespri,
Decay estimates for evolutionary equations with fractional time-diffusion, J. Evol. Equ., 19 (2019), 435-462.
doi: 10.1007/s00028-019-00482-z. |
[12] |
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. |
[13] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. {V}ol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955, https://mathscinet.ams.org/mathscinet-getitem?mr=0066496, Based, in part, on notes left by Harry Bateman. |
[14] |
F. Ferrari, {W}eyl and {M}archaud {D}erivatives: A forgotten history, Mathematics, 6 (2018), 6. https://www.mdpi.com/2227-7390/6/1/6.
doi: 10.3390/math6010006. |
[15] |
I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967, Theory of differential equations, Translated from the Russian by Meinhard E. Mayer. https://www.ams.org/books/chel/379/chel379-endmatter.pdf. |
[16] |
C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates,
The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.
doi: 10.1016/j.cnsns.2017.04.001. |
[17] |
S. L. Kalla and B. Ross, The development of functional relations by means of fractional operators, in Fractional Calculus ({G}lasgow, 1984), vol. 138 of Res. Notes in Math., Pitman, Boston, MA, 1985, 32–43. |
[18] |
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher,
Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979.
doi: 10.1007/s00208-015-1356-z. |
[19] |
J. Kemppainen, J. Siljander and R. Zacher,
Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.
doi: 10.1016/j.jde.2017.02.030. |
[20] |
J. Kemppainen and R. Zacher,
Long-time behavior of non-local in time {F}okker–{P}lanck equations via the entropy method, Math. Models Methods Appl. Sci., 29 (2019), 209-235.
doi: 10.1142/S0218202519500076. |
[21] |
Y. Luchko,
Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.
|
[22] |
Y. Luchko,
Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.
doi: 10.1016/j.jmaa.2010.08.048. |
[23] |
F. Mainardi,
On some properties of the Mittag-Leffler function {$E_\alpha(-t^\alpha)$}, completely monotone for {$t>0$} with $0 < \alpha < 1$, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2267-2278.
doi: 10.3934/dcdsb.2014.19.2267. |
[24] |
F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153–192. https://arXiv.org/pdf/cond-mat/0702419.pdf. |
[25] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, vol. 43 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2012. https://www.degruyter.com/view/product/129781. |
[26] |
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999.
![]() |
[27] |
J.-M. Roquejoffre and A. Tarfulea,
Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion, J. Math. Pures Appl. (9), 108 (2017), 399-424.
doi: 10.1016/j.matpur.2017.07.001. |
[28] |
B. Ross, The Development, Theory and Application of the Gamma-function and a Profile of Fractional-calculus, ProQuest LLC, Ann Arbor, MI, 1974, Thesis (Ph.D.)–New York University. http://gateway.proquest.com.pros.lib.unimi.it/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7417154 |
[29] |
B. Ross,
The development of fractional calculus 1695–1900, Historia Math., 4 (1977), 75-89.
doi: 10.1016/0315-0860(77)90039-8. |
[30] |
B. Ross,
Origins of fractional calculus and some applications, Internat. J. Math. Statist. Sci., 1 (1992), 21-34.
|
[31] |
J. Sánchez and V. Vergara,
Long-time behavior of bounded global solutions to systems of nonlinear integro-differential equations, Asymptot. Anal., 85 (2013), 167-178.
doi: 10.3233/ASY-131180. |
[32] |
J. Sánchez and V. Vergara,
Long-time behavior of nonlinear integro-differential evolution equations, Nonlinear Anal., 91 (2013), 20-31.
doi: 10.1016/j.na.2013.06.006. |
[33] |
T. Sandev, A. Schulz, H. Kantz and A. Iomin,
Heterogeneous diffusion in comb and fractal grid structures, Chaos Solitons Fractals, 114 (2018), 551-555.
doi: 10.1016/j.chaos.2017.04.041. |
[34] |
F. Santamaria, S. Wils, E. D. Schutter and G. J. Augustine, The diffusional properties of dendrites depend on the density of dendritic spines, Eur. J. Neurosci., 34 (2011), 561–568. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3156966/.
doi: 10.1111/j.1460-9568.2011.07785.x. |
[35] |
H. Schiessel, C. Friedrich and A. Blumen, Applications to problems in polymer physics and rheology, in Applications of Fractional Calculus in Physics, World Sci. Publ., River Edge, NJ, 2000,331–376.
doi: 10.1142/9789812817747_0007. |
[36] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with {C}aputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
[37] |
V. Vergara and R. Zacher,
A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73 (2010), 3572-3585.
doi: 10.1016/j.na.2010.07.039. |
[38] |
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. |
[39] |
R. Wong and Y.-Q. Zhao,
Exponential asymptotics of the {M}ittag-{L}effler function, Constr. Approx., 18 (2002), 355-385.
doi: 10.1007/s00365-001-0019-3. |
[40] |
R. Zacher,
Maximal regularity of type {$L_p$} for abstract parabolic {V}olterra equations, J. Evol. Equ., 5 (2005), 79-103.
doi: 10.1007/s00028-004-0161-z. |
[41] |
R. Zacher,
Weak solutions of abstract evolutionary integro-differential equations in {H}ilbert spaces, Funkcial. Ekvac., 52 (2009), 1-18.
doi: 10.1619/fesi.52.1. |
[42] |
R. Zacher, Time fractional diffusion equations: Solution concepts, regularity, and long-time behavior, in Handbook of Fractional Calculus with Applications. {V}ol. 2, De Gruyter, Berlin, 2019,159–179. https://www.degruyter.com/viewbooktoc/product/497030.
doi: 10.1515/9783110571660-008. |
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