# American Institute of Mathematical Sciences

November  2017, 16(6): 2201-2226. doi: 10.3934/cpaa.2017109

## Existence and convexity of solutions of the fractional heat equation

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72,09124 Cagliari, Italy 2 Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Viale L. Merello 92,09123 Cagliari, Italy

* Corresponding author

Received  January 2017 Revised  May 2017 Published  July 2017

We prove that the initial-value problem for the fractional heat equation admits an entire solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. The result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al.[1] for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.

Citation: Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109
##### References:
 [1] B. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with non-local diffusion, Arch. Rational Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1. [2] R. M. Blumenthal and R. H. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.  doi: 10.2307/1993291. [3] K. Bogdan, T. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Prob., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532. [4] C. Bucur and E. Valdinoci, Non-local Diffusion and Applications Springer, New York (2016). doi: 10.1007/978-3-319-28739-3. [5] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 367 (2014), 911-941.  doi: 10.1016/j.anihpc.2013.02.001. [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [7] L. Caffarelli, Non-local diffusions, drifts and games in Nonlinear partial differential equations (H. Holden and K. H. Karlsen eds. ), Springer, New York (2012). doi: 10.1007/978-3-642-25361-4_3. [8] L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math., 680 (2013), 191-233.  doi: 10.1515/crelle.2012.036. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [10] Z. Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.  doi: 10.4171/JEMS/231. [11] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [12] L. C. Evans, Partial Differential Equations 2nd edition, Graduate Studies in Mathematics 19 American Mathematical Society, Providence, Rhode Island, 2010. doi: 10.1090/gsm/019. [13] X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math., 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6. [14] A. Greco, Convex functions over the whole space locally satisfying fractional equations, Minimax Theory Appl., 2 (2017), 51-68. [15] A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14. [16] A. Iannizzotto, S. Mosconi and M. Squassina, $H^s$ versus $C^0$-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z. [17] K. Ishige, T. Kawakami and H. Michihisa, Asymptotic expansions of solutions of fractional diffusion equations, SIAM J. Math. Anal., 49 (2017), 2167-2190.  doi: 10.1137/16M1101428. [18] K. Ishige and P. Salani, A note on parabolic power concavity, Kodai Math. J., 37 (2014), 668-679.  doi: 10.2996/kmj/1414674615. [19] F. John, Partial Differential Equations fourth edition, Springer, New York (1982). doi: 10.1007/978-1-4684-9333-7. [20] T. Kulczycki, On concavity of solution of Dirichlet problem for the equation $(-Δ)^{1/2 \,} \varphi = 1$ in a convex planar region, J. Eur. Math. Soc., 19 (2017), 1361-1420.  doi: 10.4171/JEMS/695. [21] T. Kulczycki and M. Ryznar, Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc., 368 (2016), 281-318.  doi: 10.1090/tran/6333. [22] R. Musina and A. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations, 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304. [23] G. Pólya, On the zeros of an integral function represented by Fourier's integral, J. London Math. Soc., 1 (1926), 98-99.  doi: 10.1112/jlms/s1-1.1.12. [24] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5. [25] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [26] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [27] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sec. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783. [28] L. Vázquez, J. J. Trujillo and M. P. Velasco, Fractional heat equation and the second law of thermodynamics, Fract. Calc. Appl. Anal., 14 (2011), 334-342.  doi: 10.2478/s13540-011-0021-9. [29] D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.  doi: 10.2307/1990141. [30] Fractional heat equation: https://www.ma.utexas.edu/mediawiki/index.php/Fractional_heat-equation

show all references

##### References:
 [1] B. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with non-local diffusion, Arch. Rational Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1. [2] R. M. Blumenthal and R. H. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.  doi: 10.2307/1993291. [3] K. Bogdan, T. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Prob., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532. [4] C. Bucur and E. Valdinoci, Non-local Diffusion and Applications Springer, New York (2016). doi: 10.1007/978-3-319-28739-3. [5] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 367 (2014), 911-941.  doi: 10.1016/j.anihpc.2013.02.001. [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [7] L. Caffarelli, Non-local diffusions, drifts and games in Nonlinear partial differential equations (H. Holden and K. H. Karlsen eds. ), Springer, New York (2012). doi: 10.1007/978-3-642-25361-4_3. [8] L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math., 680 (2013), 191-233.  doi: 10.1515/crelle.2012.036. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [10] Z. Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.  doi: 10.4171/JEMS/231. [11] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [12] L. C. Evans, Partial Differential Equations 2nd edition, Graduate Studies in Mathematics 19 American Mathematical Society, Providence, Rhode Island, 2010. doi: 10.1090/gsm/019. [13] X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math., 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6. [14] A. Greco, Convex functions over the whole space locally satisfying fractional equations, Minimax Theory Appl., 2 (2017), 51-68. [15] A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14. [16] A. Iannizzotto, S. Mosconi and M. Squassina, $H^s$ versus $C^0$-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z. [17] K. Ishige, T. Kawakami and H. Michihisa, Asymptotic expansions of solutions of fractional diffusion equations, SIAM J. Math. Anal., 49 (2017), 2167-2190.  doi: 10.1137/16M1101428. [18] K. Ishige and P. Salani, A note on parabolic power concavity, Kodai Math. J., 37 (2014), 668-679.  doi: 10.2996/kmj/1414674615. [19] F. John, Partial Differential Equations fourth edition, Springer, New York (1982). doi: 10.1007/978-1-4684-9333-7. [20] T. Kulczycki, On concavity of solution of Dirichlet problem for the equation $(-Δ)^{1/2 \,} \varphi = 1$ in a convex planar region, J. Eur. Math. Soc., 19 (2017), 1361-1420.  doi: 10.4171/JEMS/695. [21] T. Kulczycki and M. Ryznar, Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc., 368 (2016), 281-318.  doi: 10.1090/tran/6333. [22] R. Musina and A. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations, 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304. [23] G. Pólya, On the zeros of an integral function represented by Fourier's integral, J. London Math. Soc., 1 (1926), 98-99.  doi: 10.1112/jlms/s1-1.1.12. [24] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5. [25] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [26] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032. [27] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sec. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783. [28] L. Vázquez, J. J. Trujillo and M. P. Velasco, Fractional heat equation and the second law of thermodynamics, Fract. Calc. Appl. Anal., 14 (2011), 334-342.  doi: 10.2478/s13540-011-0021-9. [29] D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.  doi: 10.2307/1990141. [30] Fractional heat equation: https://www.ma.utexas.edu/mediawiki/index.php/Fractional_heat-equation
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