Mathematical analysis of a chemical reaction with lumped temperature and strong absorption
The paper concerns existence, uniqueness and a few properties of the solution of the system \[ \partial u/\partial t-\Delta u=-\Phi g(v)f(u) \quad \text{in }\Omega\times(0,\infty), \qquad \alpha(\partial u/\partial n)=1-u \quad \text{in }\partial\Omega\times(0,\infty), \] \[ dv/dt=k(1- v)+\lambda\Phi g(v) \int_ \Omega f(u) \;\text{ in }(0,\infty), \quad u(x,0)=u_ 0(x)\geq 0, \quad v(0)=v_ 0\geq 0, \] where \(f(u)=(u^ +)^ p\) (\(0<p\)) and \(g(v)=e^{\gamma(1-{1/v})}\) if \(v>0\) and \(g(v)=0\) if \(v\leq 0\). The coefficients \(K\), \(\varphi\), \(\alpha\) and \(\gamma\) are positive while \(\lambda\) is positive (exothermic reaction), negative (endothermic reaction) or zero (isothermic reaction). Existence of a classical solution is achieved through an iterative scheme and a comparison technique (which also lead to uniqueness). In the steady state case a few properties of the dead core (i.e. the set \(\{x\); \(u(x)=0\}\)) are established. At last a few bounds, asymptotic behaviour and study of the dead core are investigated in the time dependent case.
scientific article; zbMATH DE number 62868