LOCAL STABILITY OF TRAVELLING FRONTS FOR A DAMPED WAVE EQUATION

doi:10.1016/S0252-9602(12)60195-7

Abstract

Abstract:

The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation ∂u_tt+u_t= u_xx−V ′(u) on R. The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation u_t= u_xx − V ′(u). Whereas a lot is known about the local stability of travelling fronts in parabolic systems, for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type. However, for the combustion or monostable type of V , the problem is much more complicated. In this paper, a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established. And then, the result is extended to the damped wave equation with a case of monostable pushed front.

Key words: travelling front, local stability, damped wave equation

CLC Number:

35B35

LUO Cao. LOCAL STABILITY OF TRAVELLING FRONTS FOR A DAMPED WAVE EQUATION[J].Acta mathematica scientia,Series B, 2013, 33(1): 75-83.

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References

[1] Aronson D G, Weinberger H F. Multidimensional nonlinear diffusion arising in population genetics. Adv Math, 1978, 30: 33–76

[2] Dunbar S R, Othmer H G. On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks//Nonlinear Oscillations in Biology and Chemistry. Lecture Notes in Biomath 66. Berlin: Springer, 1986: 247–289

[3] Gallay Th. Convergence to travelling waves in damped hyperbolic equations//Fiedler B, Gr¨oger K, Sprekels J. International Conference on Differential Equations, Berlin: World Scientific, 2000, (1): 787–793

[4] Gallay Th, Joly R. Global stability of travelling front for a damped wave equation with bistable nonlinearity. Ann Scient ´Ec Norm Sup, 2009, 42: 103–140

[5] Gallay Th, Raugel G. Stability of travelling waves for a damped hyperbolic equation. Z Angew Math Phys, 1997, 48: 451–479

[6] Goldstein S. On diffusioon by discontinuous movements, and on the telegraph equation. Quart J Mech Appl Math, 1951, 4: 129–156

[7] Hadeler K P. Hyperbolic travelling fronts. Proc Edinburgh Math Soc, 1988, 31: 89–97

[8] Hadeler K P. Travelling fronts for correlated random walks. Canad Appl Math Quart, 1994, 2: 27–43

[9] Hadeler K P. Reaction transport systems in biological modelling//Mathematics Inspired by Biology. Lecture Notes in Math 1714. Berlin: Springer, 1999: 95–150

[10] Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. Berlin: Springer, 1981

[11] Kac M. A stochastic model related to the telegrapher’s equation. Rocky Mountain J Math, 1974, 4: 497–509

[12] Sattinger D H. On the stability of waves of nonlinear parabolic systems. Adv Math, 1976, 22: 312–355

[13] Simon B. Schr¨odinger operators in the twentieth century. J Math Phys, 2000, 41: 3523–3555