On the Fisher‐KPP model with nonlocal nonlinear sources

Shen Bian

Mathematische Nachrichten, 2024, vol. 297, issue 1, 144-164

Abstract: The Cauchy problem considered in this paper is the following: 1 ut=Δu+uαM0−∫Rnu(x,t)dx,x∈Rn,t>0,u(x,0)=u0(x)≥0,x∈Rn,$$\begin{align} \hspace*{4pc}\left\{ \def\eqcellsep{&}\begin{array}{ll} u_t=\Delta u+u^\alpha {\left(M_0- \int _{{\mathbb {R}}^n} u(x,t)dx\right)},\quad & x \in {\mathbb {R}}^n, t>0, \\[3pt] u(x,0)=u_0(x)\ge 0,\quad & x \in {\mathbb {R}}^n, \end{array} \right.\hspace*{-4pc} \end{align}$$where M0>0,α>1,n≥3$M_0>0, \alpha >1, n \ge 3$. When the coefficient M0−∫Rnu(x,t)dx$M_0-\int _{{\mathbb {R}}^n} u(x,t) dx$ remains positive, (1) is analogous to 2 ut=Δu+uα,x∈Rn,t>0,u(x,0)=u0(x)≥0,x∈Rn.$$\begin{align} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{ll}u_t=\Delta u+u^\alpha ,\quad & x \in {\mathbb {R}}^n, t>0, \\[3pt] u(x,0)=u_0(x)\ge 0,\quad & x \in {\mathbb {R}}^n. \end{array} \right.} \hspace*{-6pc}\end{align}$$It is well known that when 1

Date: 2024
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