Global existence and asymptotic behavior of solutions to a nonlocal Fisher-KPP type problem

In this work, we consider a nonlocal Fisher-KPP reaction-diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in \(\mathbb{R}^n (n \ge 1)\), with reaction term \(u^\alpha(1-m(t))\), where \(m(t)\) is the total mass at time \(t\). When \(\alpha \ge 1\) an...

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Bibliographic Details
Published in:arXiv.org 2015-08
Main Authors: Bian, Shen, Chen, Li, Latos, Evangelos A
Format: Article
Language:English
Subjects:
Asymptotic properties
Boundary conditions
Computer simulation
Thermodynamics
Online Access:Get full text
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Summary:In this work, we consider a nonlocal Fisher-KPP reaction-diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in \(\mathbb{R}^n (n \ge 1)\), with reaction term \(u^\alpha(1-m(t))\), where \(m(t)\) is the total mass at time \(t\). When \(\alpha \ge 1\) and the initial mass is greater than or equal to one, the problem has a unique nonnegative classical solution. While if the initial mass is less than one, then the problem admits a unique global solution for \(n=1,2\) with any \(1 \le \alpha 2\) and the initial mass is less than one, our numerical results show that the solution exists globally in time and the mass tends to one as time goes to infinity.
ISSN:2331-8422