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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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##### References:
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