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

We study the long-time behavior of nonnegative solutions of the degenerate parabolic equation ut = (um)xx + (ε/n)(un)x, 0 < x < 1, t > 0, subject to the boundary conditions u(0, t) = 0, (um)x (1, t) = aup(1, t), t > 0. Here a, ε > 0 and p, n ≥ m ≥ 1. Bifurcation diagrams for the steady states are given for all cases of n, m, p > 0, and the stability or instability of each branch is obtained in the case p, n ≥ m ≥ 1. It is shown that some solutions can blow up in finite time. Generalizations replacing um by φ(u), (ε/n) un by ƒ(u), and aup by g(u) are discussed.