On the diffusion of impurities with long-range action (Q1176931)

The author considers the following nonlinear system describing the diffusion of a two-component mixture when the diffusion coefficients depend both on the local densities and their average values: \[ \begin{aligned}{\partial\beta(u(x,t))\over \partial t} & =\varphi(\| u(\cdot,t)\|, \| v(\cdot,t)\|)\Delta u(x,t) \\ {\partial v(x,t)\over \partial t} & =\psi(\| u(\cdot,t)\|, \| v(\cdot,t)\|)\Delta v(x,t), \tag{1}\end{aligned} \] under the conditions \[ \begin{aligned} u(x,0) & =u^ 0(x),\;v(x,0)=v^ 0(x),\;x\in\overline\Omega \\ u(x,t) & =v(x,t)=0,\;(x,t)\in S.\tag{2}\end{aligned} \] Here \(\|\cdot\|\) denotes the \(L_ 2(\Omega)\) norm; \(\Omega\subset R^ n\); \(S=\partial\Omega\times(0,+\infty)\). Under the condition \(\beta(\xi)=|\xi|^{m-1}\xi\), \(m>2\) and fairly general assumptions on \(\varphi\), \(\psi\), \(u^ 0\) and \(v^ 0\), the author proves the existence of a generalized solution of (1)-(2). A condition is found ensuring the strict positivity of \(u(x,t)\) everywhere inside \(\Omega\) for arbitrarily large times.