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The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations
Author(s) -
Witelski Thomas P.
Publication year - 1996
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1996973277
Subject(s) - cahn–hilliard equation , mathematics , nonlinear system , heat equation , mathematical analysis , regularization (linguistics) , singular perturbation , diffusion equation , burgers' equation , perturbation (astronomy) , convection–diffusion equation , partial differential equation , physics , economy , quantum mechanics , artificial intelligence , computer science , economics , service (business)
We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward‐backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn‐Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher‐order regularization terms uniquely determines the interface structure in these equations. It is shown that the well‐known “equal area” rule for the Cahn‐Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn‐Hilliard equation.