Abstract
We consider nearly spherically symmetric expanding fronts in the scalar bistable reaction-diffusion equation on RN. As t→∞, the front is known to look more and more like a sphere under the rescaling of the radius to unity. In this paper we prove that, if the initial state is spherically symmetric and approximated by a one-dimensional traveling wave with a sufficiently large radius, then the solution is approximated uniformly for all t≥0 without the rescaling of the radius by the one-dimensional traveling wave with the speed of V=c−(N−1)κ, where c>0 is the speed of the one-dimensional traveling wave solution and κ the mean curvature of the sphere. We further show that, if the initial state is a slightly perturbed one from the spherical front, the difference between the actual front and the expanding sphere hardly grows or decays for all t≥0, although the relative magnitude of the perturbation to the radius of the sphere decreases to zero.
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Yagisita, H. Nearly Spherically Symmetric Expanding Fronts in a Bistable Reaction-Diffusion Equation. Journal of Dynamics and Differential Equations 13, 323–353 (2001). https://doi.org/10.1023/A:1016632124792
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DOI: https://doi.org/10.1023/A:1016632124792