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

Abstract

Reaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(x,t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u$, 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 $D\left(u\right)=(u-a)(u-b)$ that is negative for $u\in (a,b)$. 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 $u=0$ and $u=1$ constant solutions, and prove for certain $a$ and $b$ 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.