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 $${\partial }_{t}u=\Delta u^m-|x{|}^{\sigma }{u}^p,(x,t)\in {\Re }^N\times (0,\infty ),N\ge 1,$$ with exponents $1<p<m$ and $\sigma >0$, are classified. Looking for solutions in the form $$u(x,t)=t^{-\alpha }f(|x|t^{\beta }),\alpha =\frac{\sigma +2}{\sigma (m-1)+2(p-1)},\beta =\frac{m-p}{\sigma (m-1)+2(p-1)},$$ it is shown that all their profiles satisfy the behavior at infinity given by $$\underset{\xi \to \infty }{\lim }{\xi }^{\sigma /(p-1)}f(\xi )={\left(\frac{1}{p-1}\right)}^{1/(p-1)},$$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f^{\prime }(0)=0$, another unique solution such that $f$ presents a \emph{dead-core}; that is, $f\equiv 0$ for $\xi \in [0,{\xi }_0]$ for some ${\xi }_0>0$, and, finally, there exists $K^{\ast }\in (0,\infty )$ such that, for any $K\in (0,K^{\ast })$, there is at least a solution such that $$\underset{\xi \to 0}{\lim }{\xi }^{-(\sigma +2)/(m-p)}f(\xi )=K.$$ 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.