×

zbMATH — the first resource for mathematics

Periodic travelling wave solutions of a curvature flow equation in the plane. (English) Zbl 1138.35035
This paper deals with the study of periodic travelling wave solutions of a curvature flow equation in the plane. The main result of this paper establishes the existence and the uniqueness of such a solution, whose graphic is a periodic ondulating line which is in a finite distance from a straight line with a prescribed inclination \(\alpha\), so that the propagation is just like that in oblique disposed striations. Two particular cases have a particular interest in this analysis. First, if \(\alpha =0\), then the periodic travelling wave solution is a horizontal straight line which travels in the \(y\)-direction with average speed \(c_0\). Next, in the case \(\alpha =\pi /2\), then there exists not-periodic travelling wave solution which travels in the -\(x\)-direction with a speed depending on the arithmetic means of two well-defined quantities.

MSC:
35K55 Nonlinear parabolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B10 Periodic solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] C. M. Brauner, P. Fife, G. Namah and C. Schmidt-Lainé, Propagation of a combustion front in a striated solid medium: a homogenization analysis, Quart. Appl. Math. 51 (1993), 467–493. · Zbl 0803.35009
[2] X. Chen and G. Namah, Wave propagation under curvature effects in a heterogeneous medium, Appl. Anal. 64 (1997), 219–233. · Zbl 0878.35009
[3] X. Chen and G. Namah, Periodic travelling wave solutions of a parabolic equation: a monotonicity result, J. Math. Anal. Appl. 275 (2002), 804–820. · Zbl 1072.35539
[4] N. Krylov and N. N. Bogoliubov, The application of methods of nonlinear mechanics to the theory of stationary oscillations, Publication 8 of the Ukrainian Academy of Science, Kiev, 1934.
[5] G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinear parabolic equations, NoDEA Nonlinear Differential Equations Appl. 4 (1997), 521–536. · Zbl 0887.35070
[6] H. Ninomiya and M. Taniguchi, Travelling curved fronts of a curvature flow with constant driving force, Free Boundary Problems: Therory and Applications I, (Chiba, 1999), 206–221, Gakuto Internat. Ser. Math. Sci. Appl. 13, Gakkotosho, 2000, Tokyo. · Zbl 0957.35124
[7] Y. Nishiura, Far-from-equilibrium dynamics, Transl. Math. Monogr. 209, Iwanami Series in Modern Mathematics, American Math. Soc., Providence, R.I., 2002. · Zbl 1013.37001
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.