Seminar on Partial Differential Equations

We consider continuous nonnegative solutions to a doubly nonlinear parabolic problem with the $p$-Laplacian with zero Dirichlet boundary conditions. For simplicity we assume that both the initial data and the reaction function are continuous and nonnegative and the reaction function does not depend on $u$. We show that for $1<p<2$ the speed of propagation is infinite in the sense that for any fixed time the solution is either everywhere positive or identically zero. In particular, if the initial data are nonzero at at least one point, then for small positive time the solution is positive in the whole domain, i.e., the strong maximum principle holds. We will also apply maximum and comparison principles to problems from turbulent filtration of natural gas in porous rock and groundwater filtration in gravel. In particular, we will focus on a model of turbulent filtration of natural gas in a porous rock due to Leibenson. This is... more