Travelling wave behaviour for a Porous-Fisher equation

Abstract

We study the propagation properties of the reaction-diffusion equation of Fisher type

with p<1<m. Taking into account that solutions of the Cauchy problem are nonunique if m+p⩾2, we prove that the minimal solutions in this case tend to propagate with certain minimal speed c∗(m, p). More precisely, if we translate any solution with a velocity ∣c∣<c∗, we get the limit in time value one, and if the initial value u(·, 0) vanishes, say, for x⩾0, the minimal solution translated with velocity c>c∗ tends to zero. Also, an interface appears for the minimal solutions, whose asymptotic velocity is c∗. This behaviour depends upon the existence of special solutions of travelling wave form. Travelling waves have been widely studied for diffusion equations related with the above. We characterize here the minimal velocity c∗ for which travelling waves exist, as an analytic function of the parameters m and p, for every m+p⩾2, by viewing it as an anomalous exponent. Some local properties of the minimal solutions and their interfaces in the case m+p⩾2 are also proved.