On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 141-155.

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C 2 classical regularity, but also the existence of discontinuous entropy traveling wave solutions.

DOI : 10.1016/j.anihpc.2012.07.001
Classification : 35K55, 35B10, 35B40, 35K57, 35K40
Mots-clés : Flux limited, Relativistic heat equation, Singular traveling waves, Nonlinear reaction–diffusion, KPP, Traveling waves, Optimal mass transportation, Entropy solutions, Complex systems, Traffic flow, Biomathematics
@article{AIHPC_2013__30_1_141_0,
     author = {Campos, Juan and Guerrero, Pilar and S\'anchez, \'Oscar and Soler, Juan},
     title = {On the analysis of traveling waves to a nonlinear flux limited reaction{\textendash}diffusion equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {141--155},
     publisher = {Elsevier},
     volume = {30},
     number = {1},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.07.001},
     mrnumber = {3011295},
     zbl = {1263.35059},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.07.001/}
}
TY  - JOUR
AU  - Campos, Juan
AU  - Guerrero, Pilar
AU  - Sánchez, Óscar
AU  - Soler, Juan
TI  - On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 141
EP  - 155
VL  - 30
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.07.001/
DO  - 10.1016/j.anihpc.2012.07.001
LA  - en
ID  - AIHPC_2013__30_1_141_0
ER  - 
%0 Journal Article
%A Campos, Juan
%A Guerrero, Pilar
%A Sánchez, Óscar
%A Soler, Juan
%T On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 141-155
%V 30
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.07.001/
%R 10.1016/j.anihpc.2012.07.001
%G en
%F AIHPC_2013__30_1_141_0
Campos, Juan; Guerrero, Pilar; Sánchez, Óscar; Soler, Juan. On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 141-155. doi : 10.1016/j.anihpc.2012.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2012.07.001/

[1] F. Andreu, J. Calvo, J.M. Mazón, J. Soler, On a nonlinear flux-limited equation arising in the transport of morphogens, J. Differential Equations 252 no. 10 (2012), 5763-5813 | MR | Zbl

[2] F. Andreu, V. Caselles, J.M. Mazón, A Fisher–Kolmogorov equation with finite speed of propagation, J. Differential Equations 248 (2010), 2528-2561 | MR | Zbl

[3] F. Andreu, V. Caselles, J.M. Mazón, Some regularity results on the relativistic heat equation, J. Differential Equations 245 (2008), 3639-3663 | MR | Zbl

[4] F. Andreu, V. Caselles, J.M. Mazón, The Cauchy problem for a strongly degenerate quasilinear equation, J. Eur. Math. Soc. (JEMS) 7 (2005), 361-393 | EuDML | MR | Zbl

[5] F. Andreu, V. Caselles, J.M. Mazón, S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal. 182 (2006), 269-297 | MR | Zbl

[6] S.V. Apte, K. Mahesh, T. Lundgren, Accounting for finite-size effects in simulations of disperse particle-laden flows, Int. J. Multiph. Flow 34 (2008), 260-271

[7] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, Partial Differential Equations and Related Topics, Lecture Notes in Math. vol. 446, Springer, New York (1975), 5-49 | Zbl

[8] D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33-76 | MR | Zbl

[9] N. Bellomo, Modelling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach, Birkhäuser, Springer, Boston (2008) | MR | Zbl

[10] N. Bellomo, J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci. 22 no. Suppl. 4 (2012) | MR | Zbl

[11] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems. I – Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), 173-213 | EuDML | MR | Zbl

[12] H. Berestycki, L. Rossi, Reaction–diffusion equations for population dynamics with forced speed, I – The case of the whole space, Discrete Contin. Dyn. Syst. 21 (2008), 41-67 | MR | Zbl

[13] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), 949-1032 | MR | Zbl

[14] H. Berestycki, F. Hamel, Generalized traveling waves for reaction–diffusion equations, Perspectives in Nonlinear Partial Differential Equations, in Honor of H. Brezis, Contemp. Math. vol. 446, Amer. Math. Soc., Providence, RI (2007) | Zbl

[15] F. Berthelin, P. Degond, M. Delitala, M. Rascle, A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal. 187 (2008), 185-220 | MR | Zbl

[16] Y. Brenier, Extended Monge–Kantorovich theory, Optimal Transportation and Applications: Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 2–8, 2001, Lecture Notes in Math. vol. 1813, Springer-Verlag, Berlin (2003), 91-122 | Zbl

[17] J. Calvo, J. Mazón, J. Soler, M. Verbeni, Qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens, Math. Models Methods Appl. Sci. 21 no. Suppl. 1 (2011), 893-937 | MR | Zbl

[18] P. Constantin, A. Kiselev, A. Oberman, L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal. 154 (2000), 53-91 | MR | Zbl

[19] J. Dolbeault, O. Sánchez, J. Soler, Asymptotic behaviour for the Vlasov–Poisson system in the stellar dynamics case, Arch. Ration. Mech. Anal. 171 (2004), 301-327 | MR | Zbl

[20] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lect. Notes Biomath. vol. 28, Springer-Verlag (1979) | MR | Zbl

[21] R.A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), 335-369 | JFM

[22] Christopher P. Grant, Theory of Ordinary Differential Equations, Brigham Young University (2008)

[23] K.P. Hadeler, F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Math. Biol. 2 (1975), 251-263 | MR | Zbl

[24] P. Hartman, Ordinary Differential Equations, Wiley, New York (1964) | MR | Zbl

[25] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Étude de lʼéquation de la diffusion avec croissance de la quantité de matiére et son application á un problḿe biologique, Bulletin Université de Etatá Moscou, Série Internationale A 1 (1937), 1-26, P. Pelcé (ed.), Dynamics of Curved Fronts, Academic Press (1988), 105-130 | Zbl

[26] A.J. Majda, P.E. Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math. 51 (1998), 1337-1348 | MR | Zbl

[27] H. Meinhardt, Models of Biological Pattern Formation, Academic Press (1982)

[28] D. Mihalas, B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, Oxford (1984) | MR | Zbl

[29] J.D. Murray, Mathematical Biology, Springer-Verlag (1996) | MR

[30] J. Rinzel, J.B. Keller, Traveling wave solutions of a nerve conduction equation, Biophys. J. 13 (1973), 1313-1336 | MR

[31] Ph. Rosenau, Tempered diffusion: A transport process with propagating front and inertial delay, Phys. Rev. A 46 (1992), 7371-7374

[32] K. Saha, D.V. Schaffer, Signal dynamics in Sonic hedgehog tissue patterning, Development 133 (2006), 889-900

[33] F. Sánchez-Garduño, P.K. Maini, Existence and uniqueness of a sharp traveling wave in degenerate non-linear diffusion Fisher–KPP equations, J. Math. Biol. 33 (1994), 163-192 | MR | Zbl

[34] D.H. Sattinger, Stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976), 312-355 | MR | Zbl

[35] A.I. Volpert, V.A. Volpert, V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. vol. 140, Amer. Math. Soc. (1994) | MR | Zbl

[36] A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B 237 (1952), 37-72 | MR

Cité par Sources :