Traveling wave solutions of a nonlinear degenerate parabolic system from petroleum engineering (Q998755)

The authors study a system of evolution PDEs motivated by real world applications. The functions involved in the system are the preassure \(p\) of a fluid flowing in a fragile porous rock (such as diatomite) and the damage parameter \(\omega\), which is a suitably averaged fraction of the rock's microcracks. In [\textit{M. Bertsch, R. Dal Passo} and \textit{C. Nitsch}, Interfaces Free Bound. 7, No. 3, 255--276 (2005; Zbl 1079.35102)], nonnegative complactly supported solutions \((\omega,p)\) were constructed, for which the supports of \(\omega\) and \(p\) coincide and never shrink. In this paper, the behavior of \(\omega\) and \(p\) near the boundary of their support is dealt with. In particular, the authors construct traveling wave solutions exhibiting jumps at this interface. The analysis involved for the proof is quite delicate, since the system possesses a strong sensibility to parameters.