Semilinear parabolic systems modelling spatially inhomogeneous exothermic reactions (Q1310856)

The authors consider reaction-diffusion systems of the form: \[ \partial_ t u-d_ 1\Delta =-uvF(\theta),\qquad \partial_ t v-d_ 2\Delta =uvf(\theta)- \mu g(v),\qquad \partial_ t \theta-d_ 3\Delta =\mu g(v) \] on a smooth bounded domain of \(\mathbb{R}^ n\) subject to homogeneous Neumann boundary conditions. The function \(f\) is given by \(f(\theta)=0\), if \(\theta\leq 0\), and \(f(\theta)= Ke^{-E/\theta}\), \(K,E\in (0,\infty)\), if \(\theta>0\), and \(g(v)\) is either equal to \(v\) or to \(v^ 2\). The authors establish the existence and uniqueness of global nonnegative solutions for nonnegative initial conditions and show uniform convergence of such solutions to spatially homogeneous steady states as \(t\to\infty\).