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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  January 2021 Early access  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 and Continuous Dynamical Systems, 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. [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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