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28 March 2024
 
  » arxiv » 1605.5506

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Convergence to traveling waves in the Fisher-Kolmogorov equation with a non-Lipschitzian reaction term
Pavel Drábek ; Peter Takáč ;
Date 18 May 2016
AbstractWe consider the semi linear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance of an advantageous gene in biology. Its non-smooth reaction function $f(u)$ allows for the introduction of travelling waves with a new profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions $u(x,t)$, $(x,t)in mathbb{R} imes mathbb{R}_+$. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave $U$. Our main result is the uniform convergence (for $xin mathbb{R}$) of every solution $u(x,t)$ of the Cauchy problem to a single traveling wave $U(x-ct + zeta)$ as $t o infty$. The speed $c$ and the travelling wave $U$ are determined uniquely by $f$, whereas the shift $zeta$ is determined by the initial data.
Source arXiv, 1605.5506
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