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 |
Keywords:
existence; uniqueness; stationary solutions; bistable Cahn-Allen diffusion equation; monotonicity; limiting properties; dynamic problemReferences:
| [1] | G. Caginalp and P. C. Fife,Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math.48, 506-518 (1988). · doi:10.1137/0148029 |
| [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 · doi:10.3792/pjaa.65.207 |
| [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 · doi:10.1007/BF01221125 |
| [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 |
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