Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Gallay, Thierry; Joly, Romain

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 103-140.

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 $u(x,t)=h(x-st)$ 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 $\left|x\right|$ 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 [38] et repose sur l’existence pour notre système d’une fonction de Lyapunov dans tout référentiel en translation uniforme.

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 $u(x,t)=h(x-st)$ 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 $\left|x\right|$, 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 [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame.

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

[2] 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 | Zbl

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

[4] 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 | Zbl

[5] 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 | Zbl

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

[7] 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 | Zbl

[8] 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 | Zbl

[9] 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 | Zbl

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

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

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

[13] 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 | Zbl

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

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

[16] 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 | Zbl

[17] 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 | Zbl

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

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

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

[21] 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 | Zbl

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

[23] 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 | Zbl

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

[25] 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 | Zbl

[26] 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 | Zbl

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

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

[29] 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

[30] 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 | Zbl

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

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

[33] 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 | Zbl

[34] 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 | Zbl

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

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

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

[38] 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 | Zbl

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

[40] 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 | Zbl

[41] 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 | Zbl

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

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

[44] 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 | Zbl