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

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
104
Issue
2
Identifiers
DOI: 10.1006/jdeq.1993.1077

Abstract

Abstract We study the long-time behavior of nonnegative solutions of the degenerate parabolic equation u t = ( u m ) xx + (ϵ/ n)( u n ) x , 0 < x < 1, t > 0, subject to the boundary conditions u(0, t) = 0, ( u m ) x (1, t) = au p (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 u m by φ( u), (ϵ/ n) u n by ƒ( u), and au p by g( u) are discussed.

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