×

zbMATH — the first resource for mathematics

The dynamics of weakly interacting fronts in an adsorbate-induced phase transition model. (English) Zbl 1193.35006
Summary: M. Hildebrand et al. [Phys. Rev. Lett. 83, 1475–1478 (1999)] proposed an adsorbate-induced phase transition model. For this model, Y. Takei et al. [Sci. Math. Jpn. 61, No. 3, 525–534 (2005; Zbl 1081.37541)] found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in \(\mathbb{R}^2\) and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates the shape of the solution in the neighborhood of the interface.
MSC:
35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35K45 Initial value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
PDF BibTeX XML Cite
Full Text: Link EuDML
References:
[1] S.-I. Ei: The motion of weakly interacting pulses in reaction-diffusion systems. J. Dynam. Differential Equations 14 (2002), 85-137. · Zbl 1007.35039
[2] S.-I. Ei and T. Ohta: Equation of motion for interacting pulse. Phys. Rev. E 50 (1994), 4672-4678.
[3] M. Eiswirth, M. Bär, and H. H. Rotermund: Spatiotemporal selforganization on isothermal catalysts. Physica D 84 (1995), 40-57.
[4] M. Funaki, M. Mimura, and T. Tsujikawa: Travelling front solutions arising in the chemotaxis-growth model. Interfaces and Free Boundaries 8 (2006), 223-245. · Zbl 1106.35119
[5] M. Hildebrand: Selbstorganisierte nanostrukturen in katakyschen oberflächenreaktionen. Ph.D. Dissertation, Mathematisch-Naturwissenschaftlichen Fakultät I, Humboldt-Universität, Berlin 1999.
[6] M. Hildebrand, M. Ipsen, H. S. Mikhailov, and G. Ertl: Localized nonequailibrium nanostructures in surface chemical reactions. New J. Phys. 5 (2003), 61.1-61.28.
[7] M. Hildebrand, M. Kuperman, H. Wio, and A. S. Mikhailov: Self-organized chemical nanoscale microreactors. Phys. Rev. Lett. 83 (1999), 1475-1478.
[8] K. Kuto and T. Tsujikawa: Pattern formation for adsorbate-induced phase transition model. RIMS Kokyuroku Bessatsu B3 (2007), 43-58. · Zbl 1221.35060
[9] A. v. Oertzen, H. H. Rotermund, A. S. Mikhailov, and G. Ertl: Standing wave patterns in the CO oxidation reaction on a Pt(110) surface: experiments and modeling. J. Phys. Chem. B 104 (2000), 3155-3178.
[10] Y. Takei, T. Tsujikawa, and A. Yagi: Numerical computations and pattern formation for adsorbate-induced phase transition model. Sci. Math. Japon. 61 (2005), 525-534. · Zbl 1081.37541
[11] Y. Takei, M. Efendiev, T. Tsujikawa, and A. Yagi: Exponential attractor for an adsorbate-induced phase transition model in non smooth domains. Osaka J. Math. 43 (2006), 215-237. · Zbl 1135.37030
[12] T. Tsujikawa: Singular limit analysis of an adsorbate-induced phase transition model. Preprint. · Zbl 1011.35031
[13] T. Tsujikawa and A. Yagi: Exponential attractor for an adsorbate-induced phase transition model. Kyushu J. Math. 56 (2002), 313-336. · Zbl 1011.35031
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.