On discontinuous travelling wave solutions for a class of hyperbolic reaction–diffusion models

•A general class of hyperbolic reaction–diffusion models is derived.•Extended Thermodynamics approach is used to obtain a symmetric hyperbolic system.•The non-existence of smooth travelling waves is proved in a region of state-space.•As example the infiltration of rain water into semi-arid environment is described.

A class of hyperbolic reaction–diffusion models is derived within the context of Extended Thermodynamics. This kind of models avoids the unphysical features concerning the instantaneous diffusive effects typical of parabolic equations and it results to be suitable for describing invasive phenomena with a well-defined boundary. Under suitable assumptions the non-existence of smooth travelling waves is proved within a region in the state-space. As an illustrative example of such a theoretical analysis, a model describing the infiltration of rain water in semi-arid environment is derived and both continuous and discontinuous travelling waves are investigated.