Premium
On singular diffusion equations with applications to self‐organized criticality
Author(s) -
Chayes J. T.,
Osher S. J.,
Ralston J. V.
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160461004
Subject(s) - mathematics , traveling wave , conservation law , scalar (mathematics) , integrable system , mathematical physics , mathematical analysis , parameterized complexity , nonlinear system , boundary value problem , physics , combinatorics , geometry , quantum mechanics
We consider solutions of the singular diffusion equation t , = ( u m −1 u x ) x , m ≦ 0, associated with the flux boundary condition lim x →−∞ ( u m −1 u x ) x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L 1 for −1 < m ≦ 0, but have non‐integrable tails for m ≦ −1. We also show that these traveling waves are the same as those for the scalar conservation law u t = −[ f ( u )] x + u xx for a particular nonlinear convection term f ( u ) = f ( u ; m , λ). © 1993 John Wiley & Sons, Inc.