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 negative diffusivity can model
aggregation, which is the preferred behaviour in some circumstances. In this
paper, we consider a nonlinear RDE with quadratic diffusivity D(u)=(uβa)(uβb) that is negative for uβ(a,b). We use a non-classical symmetry to
construct analytic receding time-dependent, colliding wave, and receding
travelling wave solutions. These solutions are initially multi-valued, 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 travelling waves are spectrally stable. Additionally,
we introduce an 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
non-symmetric diffusivity it results in a different shock position.Comment: 35 pages, 10 figure