Global existence in the critical and subcritical cases to the Fisher-KPP model with nonlocal nonlinear reaction

Abstract

The Cauchy problem considered in this paper is the following \begin{align} \left\{ \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, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} where $M_0>0, \alpha >1, n \ge 3$. When the coefficient $M_0-\int_{\mathbb{R}^n} u(x,t) dx$ remains positive, \er{nkpp0} is analogous to \begin{align} \left\{ \begin{array}{ll} u_t=\Delta u+u^\alpha,\quad & x \in \mathbb{R}^n, t>0, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} It is well known that when $1<\alpha \le 1+2/n$, the local solution of \er{fujita} blows up in finite time as long as the initial value is nontrivial. The present paper forms a contrast to \er{fujita} and shows the global existence of solutions to \er{nkpp0} for $1<\alpha \le 1+2/n$ by dealing with the mathematical challenge which is from the nonlocal term $\int_{\mathbb{R}^n} u dx$. It's proved that when $1<\alpha <1+2/n$, such a global bound is obtained for any positive $M_0$ and any non-negative initial data. While if $\alpha=1+2/n$, then the global solution does exist for sufficiently small $M_0$ and any non-negative initial data. Furthermore, the large time behavior of the global solution is also discussed for $\alpha=1+2/n$. Besides, this paper establishes the hyper-contractivity of a global solution in $L^\infty(\mathbb{R}^n)$ with $U_0 \in L^1(\mathbb{R}^n)$ for the case $\alpha=1+2/n$.