A free Boundary Problem for a Reaction-Diffusion Equation Appearing in Biology

2019 ◽  
Vol 50 (1) ◽  
pp. 95-112
Author(s):  
J. O. Takhirov
Keyword(s):  
Free Boundary ◽  
Diffusion Equation ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
A Free Boundary Problem

scholarly journals A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuki Kaneko ◽  
Hiroshi Matsuzawa ◽  
Yoshio Yamada
Keyword(s):  
Free Boundary ◽  
Diffusion Equation ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Reaction Diffusion Equation ◽  
Bistable Nonlinearity ◽  
High Space ◽  
A Free Boundary Problem

<p style='text-indent:20px;'>We study a free boundary problem of a reaction-diffusion equation <inline-formula><tex-math id="M1">\begin{document}$ u_t = \Delta u+f(u) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ t&gt;0,\ |x|&lt;h(t) $\end{document}</tex-math></inline-formula> under a radially symmetric environment in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The reaction term <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive bistable nonlinearity, which satisfies <inline-formula><tex-math id="M5">\begin{document}$ f(0) = 0 $\end{document}</tex-math></inline-formula> and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface <inline-formula><tex-math id="M6">\begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}</tex-math></inline-formula>, which expands to infinity as <inline-formula><tex-math id="M7">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>, even when the corresponding semi-wave problem does not admit solutions.</p>

Travelling wave solutions of a reaction—infiltration problem and a related free boundary problem

1994 ◽  
Vol 5 (3) ◽  
pp. 255-265 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Elena Comparini ◽  
Riccardo Ricci
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Suitable Norm ◽  
Formal Limit ◽  
A Free Boundary Problem

We consider travelling wave solutions of a reaction–diffusion system arising in a model for infiltration with changing porosity due to reaction. We show that the travelling wave solution exists, and is unique modulo translations. A small parameter ε appears in this problem. The formal limit as ε → 0 is a free boundary problem. We show that the solution for ε > 0 tends, in a suitable norm, to the solution of the formal limit.

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