Singular solutions of parabolic $p$-Laplacian with absorption

We consider, for $p\in(1,2)$ and $q>1$, the $p$-Laplacian evolution equation with absorption

$u_t = \hbox{div}\, ( \vert\nabla u\vert^{p-2} \nabla u) - u^q \quad \hbox{in } \mathbb{R}^n \times (0,\infty).$

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:

(i)

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

(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 $c$.