A porous medium equation with dominating weighted absorption: three types of self-similar solutions

Abstract: Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$\partial_tu=Δu^{m-|x|^{σu^p,}} \quad (x,t)\in\real^{N\times(0,\infty),} \quad N\geq1,$$ with exponents $1<p<m$ and $$σ\&gt;0$$, are classified. Looking for solutions in the form $$u(x,t)=t^{{-α}f(|x|t^β),} \quad α=\frac{σ+2}{σ(m-1)+2(p-1)}, \quad β=\frac{m-p}{σ(m-1)+2(p-1)},$$ it is shown that all their profiles satisfy the behavior at infinity given by $$\lim\limits_{ξ\to\infty}ξ^{{σ/(p-1)}f(ξ)=\left(\frac{1}{p-1}\right)^{1/(p-1)},}$$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)&gt;0$, $f&#39;(0)=0$, another unique solution such that $f$ presents a \emph{dead-core}; that is, $f\equiv0$ for $ξ\in[0,ξ<em>0]$ for some $ξ_0&gt;0$, and, finally, there exists $K<sup>*\in(0,\infty)$ such that, for any $K\in(0,K<sup>*)$, there is at least a solution such that $$\lim\limits</em>{ξ\to0}ξ^{{-(σ+2)/(m-p)}f(ξ)=K.}$$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.