We propose an analytical formulation of a viscosity solution resulting from a Hamilton-Jacobi type equation modeling the propagation of a flame front in a heterogeneous solid medium and where the Hamiltonian does not satisfy the usual conditions of existence and uniqueness. After constructing it, we show that it is a viscosity solution, unique up to an additive constant.

Hamilton-Jacobi equation, viscosity solutions, striated medium.

Received: October 4, 2022; Accepted: November 15, 2022; Published: December 31, 2022

How to cite this article: AKONA Tcha, EDARH-BOSSOU Toyo Koffi and d’ALMEIDA Amah Séna, A viscosity solution for a Hamilton-Jacobi type equation, Universal Journal of Mathematics and Mathematical Sciences 18(1) (2023), 17-34. http://dx.doi.org/10.17654/2277141723002

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References:

[1] Nicolas Bacaër, Minplus sectral theory and travelling front in combustion, IFAC Proceedings 34(13) (2001), 731-735.
[2] Guy Barles, Existence results for first order Hamilton-Jacobi equations, Ann. Inst. Henri Poincaré 1(5) (1984), 325-340.
[3] Patrick Bernard and Jean-Michel Roquejoffre, Convergence to time-periodic solutions in Hamilton-Jacobi equations on the circle, Comm. Partial Differential Equations 29(3-4) (2005), 457-469.
[4] Mihaï Bostan and Gawtum Namah, Remarks on bounded solutions of steady Hamilton-Jacobi equations, C. R. Acad. Sci. Paris 347(15-16) (2009), 873-878.
[5] Mihaï Bostan and Gawtum Namah, Time periodic viscosity solutions of Hamilton-Jacobi equations, Comm. Pure Appl. Anal. 6(2) (2007), 1-22.
[6] Micheal G. Crandall, Hitoshi Ishii and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, American Mathematical Society 27(1) (1992), 1-67.
[7] T. K. Edarh-Bossou, Etude de la propagation d’un front de flamme dans un milieu strié, Thèse de Doctorat, Univ Claude-Bernard Lyon I-ENS Lyon, 1993.
[8] Albert Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I. Math. 324(9) (1997), 1043-1046.
[9] Jean-Michel Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math. Pures Appl. (9) 80(1) (2001), 85-104.
[10] C. Schmidt-Lainé and Toyo Koffi Edarh-Bossou, Some results regarding an equation of Hamilton-Jacobi type, Arch. Math. (Brno) 35(3) (1999), 203-214.