The location and stability of interface solutions of an inhomogeneous parabolic problem - Norbury, J; Yeh, L (2000)
Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. Environmental effects are included through an inhomogeneous nonlinear forcing term in a parabolic equation model with a small diffusion coefficient. The small diffusivity and the form of the nonlinear forcing term yield interface solutions after a time of order one, and these interface solutions then change slowly on a longer timescale. The stability of these interface solutions is considered from both a numerical viewpoint and using asymptotic analysis, and it is found that a simple sign condition determines stability. Here the existence of a steady-state interface is determined by the zeroes of the derivative of a function H(x) ≡ In(f(x)g3(x)), and the stability of this steady state interface (which determines a patterned solution) is controlled by the sign of the second derivative of H(x) at a zero where the interface occurs. © 2001 Society for Industrial and Applied Mathematics.