# zbMATH — the first resource for mathematics

Periodic travelling wave solutions of a parabolic equation: a monotonicity result. (English) Zbl 1072.35539
Summary: This paper considers travelling wave solutions of a quasilinear parabolic equation corresponding to the propagation of a front in some striated medium: $\xi_t+R(y\sin\alpha-\xi\cos\alpha)\sqrt{1+\xi_y^2}=\nu\xi_{yy},\quad y\in\mathbb R, t\in\mathbb R.\tag{1}$ The striations, whose geometry play a crucial role on the type of solutions obtained, are supposed to be oblique and disposed in a periodic fashion. We show that the speed of such travelling waves depends in a monotonous way on the angle of inclination of the striations.

##### MSC:
 35K55 Nonlinear parabolic equations 34B15 Nonlinear boundary value problems for ordinary differential equations 35B10 Periodic solutions to PDEs
Full Text:
##### References:
 [1] Akhmerov, R.R., On periodic travelling waves of equations with viscosity coefficient of variable sign, Nonlinear anal., 13, 803-817, (1989) · Zbl 0723.76034 [2] Brauner, C.M.; Fife, P.; Namah, G.; Schmidt-Lainé, C., Propagation of a flame front in a striated solid medium: A homogenization analysis, Quart. appl. math., 51, 467-493, (1993) · Zbl 0803.35009 [3] Chen, X.; Namah, G., Wave propagation under curvature effects in a heterogeneous medium, Appl. anal., 64, 219-233, (1997) · Zbl 0878.35009 [4] Corduneanu, C., Periodic travelling waves in nonlinear diffusion models, Libertas math., 13, 187-191, (1993) · Zbl 0799.35109 [5] Iwamiya, T.; Oharu, S.; Takahashi, T., On periodic travelling wave solutions to nonlinear dispersion equations, Bull. sci. engrg. res. lab. waseda univ., 87, 84-89, (1979) · Zbl 0471.35046 [6] Namah, G., Asymptotic behaviour of the solution of a hamilton – jacobi equation, Asymptotic anal., 12, 355-370, (1996) · Zbl 0858.35022 [7] Namah, G.; Roquejoffre, J.M., Convergence to periodic fronts in a class of semilinear parabolic equations, Nonlinear differential equations appl., 4, 521-536, (1997) · Zbl 0887.35070 [8] Peletier, L.A.; Troy, W.C., Multibump periodic travelling waves in suspension bridges, Proc. roy. soc. Edinburgh sect. A, 128, 631-659, (1998) · Zbl 0909.35143 [9] Qin, T.H., Periodic solutions of nonlinear wave equations with dissipative boundary conditions, J. partial differential equations, 3, 1-12, (1990) · Zbl 0708.35053 [10] Shi, Y.; Li, T.; Qin, T.H., Periodic travelling wave solutions on nonlinear wave equations, Nonlinear anal., 35A, 917-923, (1999) · Zbl 0932.35145 [11] O. Suys, Etude de la propagation d’un front de flamme dans un milieu solide hétérogène, Ph.D. thesis, University Bordeaux 1 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.