Existence of global solutions to the Cauchy problem for some reaction-diffusion system

Abstract

We consider the Cauchy problem for the following reaction-diffusion system: $$ \begin{cases} \displaystyle \frac{\partial u_i}{\partial t} = \Delta u_i +g_i(x,t) \prod_{j=1}^m {u_j}^{p_{ij}}, & x \in {\bf R}^n, \ t > 0, \ i=1,2,\cdots,m, \\ u_i(x,0)=f_i(x) \geq 0, \ \not\equiv 0, & x \in {\bf R}^n, \ i=1,2,\cdots,m, \end{cases} $$ where $n \geq 3$, $m \geq 2$, $p_{ij} \geq 0$ $( 1 \leq i, j \leq m ),$ $ \prod_{j=1}^m {u_j}^{p_{ij}} = {u_1}^{p_{i1}} {u_2}^{p_{i2}} \cdots $ ${u_m}^{p_{im}} ,$ $ (i=1,2,\cdots,m) $ and $f_i(x)$ ($i=1,2,\cdots,m$) is a non-negative, bounded and continuous function in ${\bf R}^n$. In this paper, we show the existence of non-negative and global solutions $u_i(x,t)$ ($i=1,2,\cdots,m$) for the above Cauchy problem when $g_i(x,t)$ ($i=1,2,\cdots,m$) and $p_{ij} \geq 0$ ($1 \leq i, j \leq m$) satisfy some conditions.