Existence of global solutions to the Cauchy problem for some reaction-diffusion system. (Q636914)

The author studies the Cauchy problem \[ \frac {\partial u_i}{\partial t}=\Delta u_i + g_i(x,t)\cdot \Pi _{j=1}^m u_j^{p_{ij}},\;x\in \mathbb {R}^n,\;t>0,\;i=1,...,m, \] \( u_i(x,0)=f_i(x)\geq 0\), \( f_i(x)\not \equiv 0\), \(x \in \mathbb {R}^n\), \(i=1,...,m\), where \(n\geq 3\), \(m\geq 2\), \(p_{ij}\geq 0\), \(f_i\) are bounded and continuous, \(g_i\) are locally Hölder continuous, \( 0\leq g_i(x)\leq C(t+1)^{\theta }| x| ^\ell \) with some \(\theta \geq 0\), \(\ell >-2\), \( 1+(2+2\theta +\ell )/n < q:=\operatorname {min} \left (\Sigma _{j=1}^m p_{1j},...,\Sigma _{j=1}^m p_{mj}\right ). \) The main result states that there is \(\varepsilon >0\) such that if \(0\leq f_i(x)\leq \varepsilon (1+| x| ^{-2\lambda })/m\) with \(\lambda =(2+2\theta +\ell )/(2q-2)\) then there exist global non-negative solutions.