Semilinear parabolic systems modelling spatially inhomogeneous exothermic reactions
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 \(t\to\infty\).
scientific article; zbMATH DE number 484009