Minimal speed of fronts of reaction-convection-diffusion equations

We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of the form ut+μφ(u)ux=uxx+f(u) for positive reaction terms with f'(0)>0. The function φ(u) is continuous and vanishes at u=0. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assessment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if f''(u)/f'(0)−−−−−√+μφ'(u)<0, then the minimal speed is given by the linear value 2f'(0)−−−−−√, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case ut+μuux=uxx+u(1-u). Results are also given for the density dependent diffusion case ut+μφ(u)ux=[D(u)ux]x+f(u).