Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Wavefront solutions of scalar reaction-diffusion equations have been intensively studied for many years. There are two basic cases, typified by quadratic and cubic kinetics. An intermediate case is considered in this paper ; namely, u, = uxx + u2(1 - u). It is shown that there is a unique travelling-wave solution, with a speed 1/√2, for which the decay to zero ahead of the wave is exponential with x. Moreover, numerical evidence is presented which suggests that initial conditions with such exponential decay evolve to this travelling-wave solution, independently of the half-life of the initial decay. It is then shown that for all speeds greater than 1/√2 there is also a travelling-wave solution, but that these faster waves decay to zero algebraically, in proportion to 1/x. The numerical evidence suggests that initial conditions with such a decay rate evolve to one of these faster waves ; the particular speed depends in a simple way on the details of the initial condition. Finally, initial conditions decaying algebraically for a power Jaw other than 1/x are considered. It is shown numerically that such initial conditions evolve either to an algebraically decaying travelling wave, or in some cases to a wavefront whose shape and speed vary as a function of time. This variation is monotonic and can be quite pronounced, and the speed is a function of u as well as of time. Using a simple linearization argument, an approximate formula is derived for the wave speed which compares extremely well with the numerical results. Finally, the extension of the results to the more general case of u, = uxx + um(1 - u), with m > 1, is discussed.