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 $\alpha=1+2/n$.
Categories: math.AP