The characterization of reaction-convection-diffusion processes by travelling waves

Abstract

It has long been known that the heat equation displays infinite speed of propagation. This is to say that if the initial data are nonnegative and have nonempty compact support, the solution of an initial-value problem is positive everywhere after any infinitesimal time. However, since the nineteen-fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon. For these equations, the (generalized) solution of an initial-value problem with compactly-supported initial data will have bounded support with respect to the spatial variable at all times. In this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction-convection-diffusion equation ut = (a(u))xx + (b(u))x + c(u) is determined. The assumptions on the coefficients a, b and c are such that the classification unifies and generalizes previously-known results. The technique employed is comparision of an arbitrary solution of the equation with suitably-constructed travelling-wave solutions and subsolutions. Basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling-wave solution with bounded support. Concurrently, the search for a travelling-wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint. © 1996 Academic Press, Inc.