Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction

We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u){]}_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A^{\mathrm{\prime }}(0)=0$ and $A(u),A^{\mathrm{\prime }}(u),A^{\mathrm{\prime }\mathrm{\prime }}(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$ and of bistable type in $[s_1,1]$. We first give a trichotomy result on the asymptotic behavior of the solutions starting at compactly supported initial data, which says that, as $t\to \mathrm{\infty }$, either small-spreading (which means $u$ tends to $s_1$), or transition, or big-spreading (which means $u$ tends to $1$) happens for a solution. Then we construct the classical and sharp traveling waves (a sharp wave means a wave having a free boundary which satisfies the Darcy’s law) for the generalized degenerate diffusion equation, and then using them to characterize the spreading solution near its front.