Title: Bistable reaction equations with doubly nonlinear diffusion,
Abstract: Reaction-diffusion equations appear in biology and chemistry, and combine
linear diffusion with different kind of reaction terms. Some of them are
remarkable from the mathematical point of view, since they admit families of
travelling waves that describe the asymptotic behaviour of a larger class of
solutions $0\leq u(x,t)\leq 1$ of the problem posed in the real line. We
investigate here the existence of waves with constant propagation speed, when
the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In
the present setting we consider bistable reaction terms, which present
interesting differences w.r.t. the Fisher-KPP framework recently studied in
\cite{AA-JLV:art}. We find different families of travelling waves that are
employed to describe the wave propagation of more general solutions and to
study the stability/instability of the steady states, even when we extend the
study to several space dimensions. A similar study is performed in the critical
case that we call "pseudo-linear", i.e., when the operator is still nonlinear
but has homogeneity one. With respect to the classical model and the
"pseudo-linear" case, the travelling waves of the "slow" diffusion setting
exhibit free boundaries. \\ Finally, as a complement of \cite{AA-JLV:art}, we
study the asymptotic behaviour of more general solutions in the presence of a
"heterozygote superior" reaction function and doubly nonlinear diffusion
("slow" and "pseudo-linear").