  Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a
density field $u(\vec{x},t)$ 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 behaviour in some circumstances. In
this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u -
a)(u - b)$ that is negative for $u\in(a,b)$. We use a nonclassical symmetry to
construct analytic receding time-dependent, colliding wave, and receding
travelling wave solutions. These solutions are 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 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 non-symmetric
diffusivity it results in a different shock position.
