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Traveling waves with paraboloid like interfaces for balanced bistable dynamics. (English) Zbl 1132.35396
Summary: Cylindrically symmetric traveling waves with paraboloid like interfaces are constructed for reaction-diffusion equations with balanced bistable nonlinearities. It is shown that the interface (a level set) is asymptotically a paraboloid \(z = \frac {c}{2(n-1)|x|^{2}}\), where \((x,z)\in \mathbb R^{n}\times \mathbb R\) \((n\geqslant 2)\) is the space variable and \(c\) is the speed that the wave travels upwards in the vertical \(z\)-direction. In the two-dimensional case (i.e., \(n=1\)), the interface is asymptotically a hyperbolic cosine curve \(z=A\cosh(\mu x)\) for some positive constants \(A\) and \(\mu \).

MSC:
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
Keywords:
interface; level set
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