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January  2021, 41(1): 257-275. doi: 10.3934/dcds.2020137

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

* Corresponding author: Enrico Valdinoci

Received  September 2019 Published  February 2020

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) $.

Citation: Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

X. CabréA.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

W. E. Deming and C. G. Colcord, The minimum in the gamma function, Nature, 135 (1935), 917. doi: 10.1038/135917b0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. DipierroE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[16]

C. IonescuA. LopesD. CopotJ. 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.  Google Scholar

[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.  Google Scholar

[18]

J. KemppainenJ. SiljanderV. 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.  Google Scholar

[19]

J. KemppainenJ. 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.  Google Scholar

[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.  Google Scholar

[21]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[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.  Google Scholar

[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 Google Scholar

[29]

B. Ross, The development of fractional calculus 1695–1900, Historia Math., 4 (1977), 75-89.  doi: 10.1016/0315-0860(77)90039-8.  Google Scholar

[30]

B. Ross, Origins of fractional calculus and some applications, Internat. J. Math. Statist. Sci., 1 (1992), 21-34.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. SandevA. SchulzH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

X. CabréA.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

W. E. Deming and C. G. Colcord, The minimum in the gamma function, Nature, 135 (1935), 917. doi: 10.1038/135917b0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. DipierroE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

C. IonescuA. LopesD. CopotJ. 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.  Google Scholar

[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.  Google Scholar

[18]

J. KemppainenJ. SiljanderV. 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.  Google Scholar

[19]

J. KemppainenJ. 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.  Google Scholar

[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.  Google Scholar

[21]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[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.  Google Scholar

[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 Google Scholar

[29]

B. Ross, The development of fractional calculus 1695–1900, Historia Math., 4 (1977), 75-89.  doi: 10.1016/0315-0860(77)90039-8.  Google Scholar

[30]

B. Ross, Origins of fractional calculus and some applications, Internat. J. Math. Statist. Sci., 1 (1992), 21-34.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. SandevA. SchulzH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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