Uniqueness of the very singular solution of a degenerate parabolic equation (Q5940164)

Let \(n\geq 2\), \({{2n}\over{n+1}}<p<2\), and \(1<q<p-1+{{p}\over{n}}\). The authors establish the uniqueness of a very singular solution of the following degenerate quasilinear diffusion equation with absorption NEWLINE\[NEWLINEu_t=\Delta_p u-|u|^{q-1}u,\quad (x,t)\in Q:={\mathbb R}^n\times(0,\infty).\tag{1}NEWLINE\]NEWLINE A nonnegative continuous functions \(W\) in \({\overline Q}\setminus\{(0, 0)\}\) with \(W(x, 0)=0\) for \(x\neq 0\), \(\nabla W\in L^1_{\text{loc}}((0,\infty), W^{1, p-1}_{\text{loc}} ({\mathbb R}^n))\), and \(\int_Q W(x, t) dx\to\infty\) as \(t\to 0\) is called a very singular solution, iff \(W\) solves (1) in the distributional sense. The authors also obtain decay rates for solutions \(u(x,t)\) as \(|x|\to\infty\) assuming that the initial data satisfy \(\lim_{|x|\to\infty}|x|^{p/(q-p+1)}u(x,0)=0\).