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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 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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