Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms

This paper deals with the appearance of monotone bounded travelling wave solutions for a parabolic reaction–diffusion equation which frequently meets both in chemical and biological systems. In particular, we prove the existence of monotone front type solutions for any wave speed c ≥ c∗ and give an estimate for the threshold value c∗. Our model takes into account both a density dependent diffusion term and a non–linear convection effect. Moreover, we do not require the main non–linearity g to be a regular C1 function; in particular we are able to treat both the case when g′(0) = 0, giving rise to a degenerate equilibrium point in the phase plane, and the singular case when g′(0) = +∞. Our results generalize previous ones due to Aronson and Weinberger [Adv. Math. 30 (1978), pp. 33 – 76], Gibbs and Murray (see Murray [Mathematical Biology, Springer–Verlag, Berlin, 1993]) and McCabe, Leach and Needham [SIAM J. Appl. Math. 59 (1998), pp. 870 – 899]. Finally, we obtain our conclusions by means of a comparison–type technique which was introduced and developed in this framework in a recent paper by the same authors.