Stability and Instability for Solutions of the Convective Porous Medium Equation with a Nonlinear Forcing at the Boundary, II

AbstractWe continue our study of the long-time behavior of nonnegative solutions for the degenerate parabolic equation ut = (um)xx + (ϵ/n)(un)x, 0 < x < 1, t > 0, subject to nonlinear boundary conditions. Here we consider solutions of this equation subject to the boundary conditions. Here we consider solutions of this equation subject to the boundary conditions −(um)x (0, t) = aup(0, t), u(1, t) = 0, t > 0. As in Part I (J. R. Anderson, J. Differential Equations, to appear), we give the bifurcation diagrams for the stationary solutions. Due to the existence of singular stationary states, some of these diagrams are three dimensional. We also examine the stability properties of these states when n, p ≥ m ≥ 1