Asymptotic results for injection of reactive solutes from a three-dimensional well (Q5945097)

The authors consider the following nonlinear initial-boundary value problem: NEWLINE\[NEWLINE \begin{cases} \beta(u)_t + \text{div} F = 0\quad\text{in}\quad\Omega_\varepsilon\times\mathbb{R}_+,\cr F\cdot \mathbf{e}_r = u_e \mathbf{q}\cdot \mathbf{e}_r \quad\text{on}\quad \partial\Omega_\varepsilon\times\mathbb{R}_+,\cr u(\cdot, 0) = u_0 (\cdot)\quad\text{in}\quad\Omega_\varepsilon.\end{cases} \tag \(P_\varepsilon\) NEWLINE\]NEWLINE Here, \(\Omega_\varepsilon = \{x\in \mathbb{R}^3 : |x |>\varepsilon\}\), \(F = \mathbf{q}u - \nabla u\), and the nonlinear term \(\beta(u)\) generally takes the form \(\beta(u) = u +\psi (u)\). The main result of this article is the investigation of the behaviour of the weak solution of problem \((P_\varepsilon)\) under convergence as \(\varepsilon\to 0\).