Positive solutions of the porous medium equation with hysteresis (Q1399304)

Equations which describe water flows in unsaturated porous media and account for a simple form capillary hysteresis in relation between saturation \(\theta\) and pressure \(p\) are considered. The problem is described by the equations \[ \dot\theta=\Delta p,\quad p\in (1+k\text{ sign } \dot\theta)\theta^m, \quad 0<k<1, m>0. \] Existence and uniqueness of solutions with given initial data are established. For a given profile of saturation the relation \(p\in (1+k\text{ sign} \Delta p)\theta^m\) supplemented by Neumann boundary conditions is a boundary value problem with solution \(p=P(\theta)\). Then the original initial value problem can be written as an evolutionary equation: \(\dot\theta=\Delta P(\theta)\). It is proved this for any nonnegative \(\theta(0)\) that equation has a unique bounded solution \(\theta(t)\) which is continuous in \(t\) and has a strong derivative \(\dot\theta\) for almost all \(t>0\) in \(L_{1+\varepsilon}\) \((\varepsilon>0)\).