×

zbMATH — the first resource for mathematics

Saddle solutions of the bistable diffusion equation. (English) Zbl 0764.35048
Summary: Stationary solutions of the bistable Cahn-Allen diffusion equation in the plane are constructed, which are positive in quadrants 1 and 3 and negative in the other two quadrants. They are unique and obey certain monotonicity and limiting properties. Solutions of the dynamic problem near this stationary solution generally split the cross-shaped nullset into two disconnected parts which remain bounded away from each other.

MSC:
35K55 Nonlinear parabolic equations
35B20 Perturbations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDF BibTeX Cite
Full Text: DOI
References:
[1] G. Caginalp and P. C. Fife,Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math.48, 506-518 (1988).
[2] Xinfu Chen,Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, in press. · Zbl 0765.35024
[3] Y-G. Chen, Y. Giga, and S. Goto,Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Academy65, 207-210 (1989). · Zbl 0735.35082
[4] L. C. Evans, H. M. Soner and P. E. Souganidis,Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., in press. · Zbl 0801.35045
[5] G. Barles, H. M. Soner, and P. E. Souganidis, in preparation.
[6] L. C. Evans and J. Spruck,Motion of level set by mean curvature I, J. Diff. Geometry,33, 635-681 (1991). · Zbl 0726.53029
[7] L. C. Evans and J. Spruck,Motion of level set by mean curvature II, preprint. · Zbl 0776.53005
[8] P. C. Fife,Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics #53, Soc. Ind. Appl. Math., Philadelphia (1988).
[9] B. Gidas, Wei-Ming Ni, and L. Nirenberg,Symmetry and related properties via the maximum principle, Commun. Math. Phys.68, 209-243 (1979). · Zbl 0425.35020
[10] P. de Mottoni and M. Schatzman,Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., in press. · Zbl 0797.35077
[11] P. de Mottoni and M. Schatzman,Development of interfaces in R n , Proc. Royal Soc. Edinburgh116A, 207-220 (1990). · Zbl 0725.35009
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.