Analytic shock‐fronted solutions to a reaction–diffusion equation with negative diffusivity

Abstract

AbstractReaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field according to diffusion and net local changes. Usually, the diffusivity is positive for all values of , which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity that is negative for . We use a nonclassical symmetry to construct analytic receding time‐dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, and we convert them to single‐valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan‐like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the and constant solutions, and prove for certain and that receding traveling waves are spectrally stable. In addition, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well‐known equal‐area rule, but for nonsymmetric diffusivity it results in a different shock position.