# American Institute of Mathematical Sciences

August  2021, 41(8): 3555-3577. doi: 10.3934/dcds.2021007

## Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative

 1 Université de Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France 2 Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, Chile 3 Departamento de Matemática, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador

* Corresponding author: Erwin Topp

Received  February 2020 Published  August 2021 Early access  January 2021

Fund Project: Part of this work was done during a visit of E.T. and M.Y. to the Institut de Recherche Math´ematique de Rennes. They acknowledge the hospitality of the Institut

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $\alpha \in (0, 1)$ cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case $\alpha = 1$, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

Citation: Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. [2] Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, volume 2074 of Lecture Notes in Mathematics, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. doi: 10.1007/978-3-642-36433-4. [3] M Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z. [4] G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869. [5] G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. (JEMS), 13 (2011), 1-26.  doi: 10.4171/JEMS/242. [6] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007. [7] G. Barles, H. Ishii and H. Mitake, A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations, Bull. Math. Sci., 3 (2013), 363-388.  doi: 10.1007/s13373-013-0036-0. [8] F. Camilli, R. De Maio and E. Lacomini, A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time derivative, J. Math. Anal. Appl., 477 (2019), 1019-1032.  doi: 10.1016/j.jmaa.2019.04.069. [9] A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.  doi: 10.1137/050621955. [10] K. Diethelm, The Analysis of Fractional Differential Equations, volume 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-14574-2. [11] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990. [12] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4. [13] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6. [14] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.  doi: 10.1007/s00526-004-0271-z. [15] Y. Feng, L. Li, J. G. Liu and X. Xu, Continuous and discrete one dimensional autonomous fractional ODEs, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3109-3135.  doi: 10.3934/dcdsb.2017210. [16] Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Partial Differential Equations, 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880. [17] R. Gorenflo, A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [18] H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations for large time and related topics, In ICIAM 07—6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 193–217. [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006. [20] P.-L. Lions, B. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi Equations, Unpublished, 1986. [21] G. Namah and J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451. [22] T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 23, 39 pp. doi: 10.1007/s00030-018-0513-y. [23] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058. [24] E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. [2] Y. Achdou, G. Barles, H. Ishii and G. L. Litvinov, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, volume 2074 of Lecture Notes in Mathematics, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. doi: 10.1007/978-3-642-36433-4. [3] M Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z. [4] G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869. [5] G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. (JEMS), 13 (2011), 1-26.  doi: 10.4171/JEMS/242. [6] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007. [7] G. Barles, H. Ishii and H. Mitake, A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations, Bull. Math. Sci., 3 (2013), 363-388.  doi: 10.1007/s13373-013-0036-0. [8] F. Camilli, R. De Maio and E. Lacomini, A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time derivative, J. Math. Anal. Appl., 477 (2019), 1019-1032.  doi: 10.1016/j.jmaa.2019.04.069. [9] A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.  doi: 10.1137/050621955. [10] K. Diethelm, The Analysis of Fractional Differential Equations, volume 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-14574-2. [11] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990. [12] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.  doi: 10.1016/S0764-4442(98)80144-4. [13] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6. [14] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.  doi: 10.1007/s00526-004-0271-z. [15] Y. Feng, L. Li, J. G. Liu and X. Xu, Continuous and discrete one dimensional autonomous fractional ODEs, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3109-3135.  doi: 10.3934/dcdsb.2017210. [16] Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Partial Differential Equations, 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880. [17] R. Gorenflo, A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [18] H. Ishii, Asymptotic solutions of Hamilton-Jacobi equations for large time and related topics, In ICIAM 07—6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 193–217. [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006. [20] P.-L. Lions, B. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi Equations, Unpublished, 1986. [21] G. Namah and J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451. [22] T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Paper No. 23, 39 pp. doi: 10.1007/s00030-018-0513-y. [23] T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab., 19 (2014), 1-25.  doi: 10.1214/EJP.v19-3058. [24] E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024.
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