Partially overlapping travelling waves in a parabolic-hyperbolic system

Abstract

We study the existence of travelling wave solutions of a one-dimensional parabolic-hyperbolic system for $ u(x, t) $ and $ v(x, t) $, which arises as a model for contact inhibition of cell growth. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and strongly parameter-dependent. In the present paper we consider a parameter regime where the minimal wave speed is positive. We show that there exists a branch of travelling wave solutions for wave speeds which are larger than the minimal one. But the main result is more surprising: for certain values of the parameters the travelling wave with minimal wave speed is not segregated (a solution is called segregated if the product $ uv $ vanishes almost everywhere) and in that case there exists a second branch of 'partially overlapping' travelling wave solutions for speeds between the minimal one and that of the (unique) segregated travelling wave.