Global existence for coupled reaction-diffusion systems (Q1588413)

The following initial-boundary value problem for a reaction-diffusion system is considered in the paper: \[ \begin{aligned} &\frac {\partial u}{\partial t}-a\Delta u=f(x,t,u,v) \quad \text{in } \{x\in\Omega,\;t\in (0,\infty)\}\\ &\frac {\partial v}{\partial t}-c\Delta u-d\Delta v=g(x,t,u,v) \quad \text{in } \{x\in\Omega,\;t\in (0,\infty)\}\\&u=v=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in (0,\infty)\}\\ &v(x,0)=v_0(x),\;u(x,0)=u_0(x), \;x\in\Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with a regular boundary, \(a>0\), \(d>0\), \(c\in\mathbb{R}\). It is proved that the system has a unique global classical solution if \(f\) and \(g\) satisfy the following conditions: \(f\), \(g\) are measurable locally Lipschitz continuous in \(u,v\); \((\text{sign }u) f(x,t,u,v)\leq 0\) for all \(u,v\) and a.e. \(x,t\); there exists an \(\alpha> 0\) such that \((\text{sign }u) f(x,t,u,v)+\alpha(\text{sign }v) g(x,t,u,v)\leq 0\) for all \(u,v\) and a.e. \(x,t\); there exist constants \(L(r)\), \(M(r)>0\), \(\sigma\geq 1\) such that \[ |f(x,t,u,v)|+|g(x,t,u,v)|\leq L(r)|v|^{\sigma}+M(r) \] for all \(u,v\) with \(|u|\leq r\) and a.e. \(x,t\). Some examples are discussed in the article.