Dead cores in a convection-diffusion problem (Q1340538)

A model for fiber spinning is considered. It involves the compressible gas flow through a cylindrical multifilament bundle of parallel fibers. The volume occupied by the bundle is modelled as a porous medium with anisotropic permeability, and the flow is governed by Darcy's law. The equation for the steady density distribution for given parameters and prescribed pressure on the boundary is generalized to \(-\Delta u+ \lambda\vec a\cdot \nabla (u^p)= 0\) with \(0< p< 1\), \(u\geq 0\), and \(u= h\) on the boundary of the domain. The density is \(\rho= u^p\). The existence of a unique solution for this problem is established in \(\mathbb{R}^N\) \((N\geq 1)\). Moreover, it is proven that there exist nonempty interior dead core regions in which \(u= 0\), if some threshold value of the parameter \(\lambda\) is exceeded. If \(h= 0\) on some part of the boundary of the domain and \(\vec a\) has a nonzero component directed into the interior of the domain, then the dead core will invade into the domain for sufficiently high values of \(\lambda\).