Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure

Risler, Emmanuel

doi:10.1016/j.anihpc.2006.12.005

Risler, Emmanuel

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 2, pp. 381-424.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2006.12.005

@article{AIHPC_2008__25_2_381_0,
     author = {Risler, Emmanuel},
     title = {Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {381--424},
     publisher = {Elsevier},
     volume = {25},
     number = {2},
     year = {2008},
     doi = {10.1016/j.anihpc.2006.12.005},
     mrnumber = {2400108},
     zbl = {1152.35047},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2006.12.005/}
}

TY  - JOUR
AU  - Risler, Emmanuel
TI  - Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2008
SP  - 381
EP  - 424
VL  - 25
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2006.12.005/
DO  - 10.1016/j.anihpc.2006.12.005
LA  - en
ID  - AIHPC_2008__25_2_381_0
ER  -

%0 Journal Article
%A Risler, Emmanuel
%T Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
%J Annales de l'I.H.P. Analyse non linéaire
%D 2008
%P 381-424
%V 25
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2006.12.005/
%R 10.1016/j.anihpc.2006.12.005
%G en
%F AIHPC_2008__25_2_381_0

Risler, Emmanuel. Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 2, pp. 381-424. doi : 10.1016/j.anihpc.2006.12.005. http://www.numdam.org/articles/10.1016/j.anihpc.2006.12.005/

Bibliographie
Cité par

[1] A. Ambrosetti, M.L. Bertotti, Homoclinics for second order conservative systems, in: M. Miranda (Ed.), PDE's and Related Subjects, Trento, Italy, 1990, in: Pitman Res. Notes Math. Ser., vol. 269, 1992, pp. 21-37. | MR | Zbl

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

[3] Berestycki H., Larrrouturou B., Lions P.L., Multi-dimensional traveling wave solutions of a flame propagation model, Arch. Rat. Mech. Anal. 111 (1990) 33-49. | MR | Zbl

[4] Chen X., Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997) 125-160. | MR | Zbl

[5] Feireisl E., Bounded, locally compact global attractors for semilinear damped wave equations on $R^{n}$ , J. Diff. Int. Eq. 9 (1996) 1147-1156. | MR | Zbl

[6] Fife P., Long Time Behavior of Solutions of Bistable Nonlinear Diffusion Equations, Arch. Rat. Mech. Anal. 70 (1979) 31-46. | MR | Zbl

[7] Fife P., Mcleod J.B., The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335-361. | MR | Zbl

[8] Fife P., Mcleod J.B., A phase plane discussion of convergence to traveling fronts for nonlinear diffusion, Arch. Rat. Mech. Anal. 75 (1981) 281-314. | MR | Zbl

[9] Gallay Th., Convergence to traveling waves in damped hyperbolic equations, in: Fiedler B., Gröger K., Sprekels J. (Eds.), International Conference on Differential Equations, vol. 1, Berlin 1999, World Scientific, 2000, pp. 787-793. | MR | Zbl

[10] Th. Gallay, R. Joly, Global stability of travelling fronts for a damped wave equation with bistable nonlinearity, Preprint. | Numdam | MR | Zbl

[11] Gallay Th., Risler E., A variational proof of global stability for bistable traveling waves, Diff. Int. Equ. 20 (8) (2007) 901-926. | MR

[12] Gallay Th., Slijepcevic̀ S., Energy flow in extended gradient partial differential equations, J. Dyn. Diff. Equ. 13 (2001) 4. | MR | Zbl

[13] Ginibre J., Velo G., The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, II. Contraction methods, Comm. Math. Phys. 187 (1997) 45-79. | MR | Zbl

[14] Heinze S., A variational approach to traveling waves, Technical Report 85, Max Planck Institute for Mathematical Sciences, Leipzig, 2001.

[15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, 1981. | MR | Zbl

[16] Jendoubi M.A., Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Diff. Equ. 144 (1998) 302-312. | MR | Zbl

[17] Kato T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58 (1975) 181-205. | MR | Zbl

[18] Kelley A., The stable, center stable, center, center unstable and unstable manifolds, J. Diff. Equ. 3 (1967) 546-570. | MR | Zbl

[19] Kolmogorov A.N., Petrovskii I.G., Piskunov N.S., A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskovo Gos. Univ. 17 (1937) 1-72.

[20] Mielke A., The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity 10 (1997) 199-222. | MR | Zbl

[21] Mielke A., Schneider G., Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity 8 (1995) 743-768. | MR | Zbl

[22] Muratov C.B., A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Disc. Cont. Dyn. Syst. Ser. B 4 (2004) 867-892. | MR | Zbl

[23] Ogiwara T., Matano H., Monotonicity and convergence results in order preserving systems in the presence of symmetry, Disc. Cont. Dyn. Syst. 5 (1999) 1-34. | MR | Zbl

[24] Ogiwara T., Matano H., Stability analysis in order-preserving systems in the presence of symmetry, Proc. Roy. Soc. Edinburgh Sect. A 129 (2) (1999) 395-438. | MR | Zbl

[25] E. Risler, A global relaxation result for bistable solutions of spatially extended gradient-like systems in one unbounded spatial dimension, in preparation.

[26] E. Risler, Global behavior of bistable solutions of nonlinear parabolic systems with a gradient structure, in preparation.

[27] Roquejoffre J.-M., Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincare Anal. Non Lineaire 14 (1997) 499-552. | Numdam | MR | Zbl

[28] Vega J.-M., Multidimensional traveling fronts in a model from combustion theory and related problems, Diff. Int. Eq. 6 (1993) 131-155. | MR | Zbl

[29] Volpert A.I., Volpert V.A., Volpert V.A., Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140, AMS, Providence, RI, 1994. | MR | Zbl

[30] Xin X., Existence and uniqueness of traveling waves in a reaction-diffusion equation with combustion nonlinearity, Indiana Univ. Math. J. 40 (3) (1991) 985-1008. | MR | Zbl

Cité par Sources :