Differential and Integral Equations

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

Munemitsu Hirose

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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.

Article information

Differential Integral Equations, Volume 23, Number 7/8 (2010), 671-684.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K45: Initial value problems for second-order parabolic systems 35K57: Reaction-diffusion equations


Hirose, Munemitsu. Existence of global solutions to the Cauchy problem for some reaction-diffusion system. Differential Integral Equations 23 (2010), no. 7/8, 671--684. https://projecteuclid.org/euclid.die/1356019190

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