Travelling wave profiles in some models with nonlinear diffusion

We study some properties of the monotone solutions of the boundary value problem(P(u′))′-cu′+f(u)=0,u(-∞)=0,u(+∞)=1,where f   is a continuous function, positive in (0,1)(0,1) and taking the value zero at 0 and 1, and P   may be an increasing homeomorphism of [0,1)[0,1) or [0,+∞)[0,+∞) onto [0,+∞)[0,+∞). This problem arises when we look for travelling waves for the reaction diffusion equation∂u∂t=∂∂xP∂u∂x+f(u)with the parameter c representing the wave speed.A possible model for the nonlinear diffusion is the relativistic curvature operator P(v)=v1-v2.The same ideas apply when P is given by the one-dimensional p  -Laplacian P(v)=|v|p-2vP(v)=vp-2v. In this case, an advection term is also considered.We show that, as for the classical Fisher–Kolmogorov–Petrovski–Piskounov equations, there is an interval of admissible speeds [c∗,+∞)[c∗,+∞) and we give characterisations of the critical speed c∗c∗. We also present some examples of exact solutions.