Singular solutions of parabolic $p$-Laplacian with absorption

Abstract:

We consider, for $p\in (1,2)$ and $q>1$, the $p$-Laplacian evolution equation with absorption \[ u_t = \operatorname {div} ( |\nabla u|^{p-2} \nabla u) - u^q \quad \mathrm {in}\ \mathbb {R}^n \times (0,\infty ).\] We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in $\mathbb {R}^n\times [0,\infty )\setminus \{(0,0)\}$, and satisfy $u(x,0)=0$ for all $x\not =0$. We prove the following:

1. [(i)] When $q\geq p-1+p/n$, there does not exist any such singular solution.

2. [(ii)] When $q<p-1+p/n$, there exists, for every $c>0$, a unique singular solution $u=u_c$ that satisfies $\int _{\mathbb {R}^n}u(\cdot ,t)\to c$ as $t\searrow 0$.

Also, $u_c\nearrow u_\infty$ as $c\nearrow \infty$, where $u_\infty$ is a singular solution that satisfies $\int _{\mathbb {R}^n} u_\infty (\cdot ,t) \to \infty$ as $t\searrow 0$. Furthermore, any singular solution is either $u_\infty$ or $u_c$ for some finite positive $c$.