Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 1, p. 103-140

We consider the damped wave equation $\alpha {u}_{tt}+{u}_{t}={u}_{xx}-{V}^{\text{'}}\left(u\right)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u\left(x,t\right)=h\left(x-st\right)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$. We show that, if the initial data are sufficiently close to the profile of a front for large $|x|$, the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t\to +\infty$. The proof of this global stability result is inspired by a recent work of E. Risler  and relies on the fact that our system has a Lyapunov function in any Galilean frame.

Nous étudions l’équation hyperbolique amortie $\alpha {u}_{tt}+{u}_{t}={u}_{xx}-{V}^{\text{'}}\left(u\right)$ sur la droite réelle, où $V$ est un potentiel bistable. Cette équation possède des ondes progressives de la forme $u\left(x,t\right)=h\left(x-st\right)$ qui décrivent le mouvement d’une interface séparant deux états d’équilibre du système, dont l’un est le minimum global de $V$. Nous montrons que, si les données initiales sont suffisamment proches du profil du front pour $|x|$ grand, alors la solution de l’équation hyperbolique amortie converge uniformément sur $ℝ$ vers une onde progressive lorsque $t\to +\infty$. La démonstration de ce résultat de stabilité globale s’inspire d’un travail récent de E. Risler  et repose sur l’existence pour notre système d’une fonction de Lyapunov dans tout référentiel en translation uniforme.

DOI : https://doi.org/10.24033/asens.2091
Classification:  35B35,  35B40,  37L15,  37L7
Keywords: travelling front, global stability, damped wave equation, Lyapunov function
@article{ASENS_2009_4_42_1_103_0,
author = {Gallay, Thierry and Joly, Romain},
title = {Global stability of travelling fronts for a damped wave equation with bistable nonlinearity},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 42},
number = {1},
year = {2009},
pages = {103-140},
doi = {10.24033/asens.2091},
zbl = {1169.35041},
mrnumber = {2518894},
language = {en},
url = {http://www.numdam.org/item/ASENS_2009_4_42_1_103_0}
}

Gallay, Thierry; Joly, Romain. Global stability of travelling fronts for a damped wave equation with bistable nonlinearity. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 1, pp. 103-140. doi : 10.24033/asens.2091. http://www.numdam.org/item/ASENS_2009_4_42_1_103_0/

 D. G. Aronson & H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), 33-76. | MR 511740 | Zbl 0407.92014

 J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa & T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci. 14 (2004), 253-293. | MR 2040897 | Zbl 1058.35076

 E. A. Coddington & N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., 1955. | MR 69338 | Zbl 0064.33002

 S. R. Dunbar & H. G. Othmer, On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in Nonlinear oscillations in biology and chemistry (Salt Lake City, Utah, 1985), Lecture Notes in Biomath. 66, Springer, 1986, 274-289. | MR 853189 | Zbl 0592.92003

 M. A. Efendiev & S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001), 625-688. | MR 1815444 | Zbl 1041.35016

 E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on ${𝐑}^{N}$, Differential Integral Equations 9 (1996), 1147-1156. | MR 1392099 | Zbl 0858.35084

 P. C. Fife & J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal. 65 (1977), 335-361. | MR 442480 | Zbl 0361.35035

 P. C. Fife & J. B. Mcleod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1980), 281-314. | MR 607901 | Zbl 0459.35044

 T. Gallay, Convergence to travelling waves in damped hyperbolic equations, in International Conference on Differential Equations (Berlin, 1999), World Sci. Publ., 2000, 787-793. | MR 1870237 | Zbl 0969.35094

 T. Gallay & G. Raugel, Stability of travelling waves for a damped hyperbolic equation, Z. Angew. Math. Phys. 48 (1997), 451-479. | MR 1460261 | Zbl 0877.35021

 T. Gallay & G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations 150 (1998), 42-97. | MR 1660266 | Zbl 0913.35086

 T. Gallay & G. Raugel, Scaling variables and stability of hyperbolic fronts, SIAM J. Math. Anal. 32 (2000), 1-29. | MR 1766519 | Zbl 0963.35128

 T. Gallay & G. Raugel, Stability of propagating fronts in damped hyperbolic equations, in Partial differential equations (Praha, 1998), Chapman & Hall Notes Math. 406, 2000, 130-146. | MR 1713881 | Zbl 0931.35103

 T. Gallay & E. Risler, A variational proof of global stability for bistable travelling waves, Differential Integral Equations 20 (2007), 901-926. | MR 2339843 | Zbl 1212.35210

 T. Gallay & S. Slijepčević, Energy flow in formally gradient partial differential equations on unbounded domains, J. Dynam. Differential Equations 13 (2001), 757-789. | MR 1860285 | Zbl 1003.35085

 J. Ginibre & G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Phys. D 95 (1996), 191-228. | MR 1406282 | Zbl 0889.35045

 J. Ginibre & G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys. 187 (1997), 45-79. | MR 1463822 | Zbl 0889.35046

 S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951), 129-156. | MR 47963 | Zbl 0045.08102

 K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinburgh Math. Soc. 31 (1988), 89-97. | MR 930018 | Zbl 0726.35057

 K. P. Hadeler, Travelling fronts for correlated random walks, Canad. Appl. Math. Quart. 2 (1994), 27-43. | MR 1271437 | Zbl 0802.60065

 K. P. Hadeler, Reaction transport systems in biological modelling, in Mathematics inspired by biology (Martina Franca, 1997), Lecture Notes in Math. 1714, Springer, 1999, 95-150. | MR 1737306 | Zbl 1002.92506

 D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, 1981. | MR 610244 | Zbl 0456.35001

 R. Ikehata, K. Nishihara & H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations 226 (2006), 1-29. | MR 2232427 | Zbl 1116.35094

 M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math. 4 (1974), 497-509. | MR 510166 | Zbl 0314.60052

 J. I. Kanelʼ, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (1962), 245-288. | MR 157130 | Zbl 0173.12801

 J. I. Kanelʼ, Stabilization of the solutions of the equations of combustion theory with finite initial functions, Mat. Sb. (N.S.) 65 (1964), 398-413. | MR 177209 | Zbl 0168.36301

 G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math. 143 (2000), 175-197. | MR 1813366 | Zbl 0964.35022

 T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), 181-205. | MR 390516 | Zbl 0343.35056

 A. N. Kolmogorov, I. G. Petrovskii & N. S. Piskunov, Étude de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull. 1 (1937), 1-25. | Zbl 0018.32106

 Y. Maekawa & Y. Terasawa, The Navier-Stokes equations with initial data in uniformly local ${L}^{p}$ spaces, Differential Integral Equations 19 (2006), 369-400. | MR 2215625 | Zbl 1212.35350

 J. Matos & P. Souplet, Universal blow-up rates for a semilinear heat equation and applications, Adv. Differential Equations 8 (2003), 615-639. | MR 1972493 | Zbl 1028.35065

 A. Mielke & G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity 8 (1995), 743-768. | MR 1355041 | Zbl 0833.35016

 C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete & Contin. Dyn. Syst. 4 (2004), 867-892. | MR 2082914 | Zbl 1069.35031

 K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996), 171-188. | MR 1419010 | Zbl 0866.35066

 K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan 58 (2006), 805-836. | MR 2254412 | Zbl 1110.35047

 A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44, Springer, 1983. | MR 710486 | Zbl 0516.47023

 M. H. Protter & H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., 1967. | MR 219861 | Zbl 0153.13602

 E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 381-424. | Numdam | MR 2400108 | Zbl 1152.35047

 J.-M. Roquejoffre, Convergence to travelling waves for solutions of a class of semilinear parabolic equations, J. Differential Equations 108 (1994), 262-295. | MR 1270581 | Zbl 0806.35093

 J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 499-552. | Numdam | MR 1464532 | Zbl 0884.35013

 J.-M. Roquejoffre, D. Terman & V. A. Volpert, Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems, SIAM J. Math. Anal. 27 (1996), 1261-1269. | MR 1402439 | Zbl 0861.35013

 D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), 312-355. | MR 435602 | Zbl 0344.35051

 B. Simon, Schrödinger operators in the twentieth century, J. Math. Phys. 41 (2000), 3523-3555. | MR 1768631 | Zbl 0981.81025

 A. I. Volpert, Vi. A. Volpert & Vl. A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs 140, Amer. Math. Soc., 1994. | MR 1297766 | Zbl 1001.35060