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The existence of nonnegative radially symmetric eternal solutions ofexponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\betat};\beta)$ is investigated for the singular diffusion equation with criticalgradient absorption \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad\;\;\hbox{in}\;\; (0,\infty)\times\real^N where $2N/(N+1) < p < 2$. Suchsolutions are shown to exist only if the parameter $\beta$ ranges in a boundedinterval $(0,\beta_*]$ which is in sharp contrast with well-known singulardiffusion equations such as $\partial_{t}\phi-\Delta_{p} \phi=0$ when$p=2N/(N+1)$ or the porous medium equation $\partial_{t}\phi-\Delta\phi^m=0$when $m=(N-2)/N$. Moreover, the profile $f(r;\beta)$ decays to zero as$r\to\infty$ in a faster way for $\beta=\beta_*$ than for $\beta\in(0,\beta_*)$ but the algebraic leading order is the same in both cases. Infact, for large $r$, $f(r;\beta_*)$ decays as $r^{-p/(2-p)}$ while $f(r;\beta)$behaves as $(\log r)^{2/(2-p)} r^{-p/(2-p)}$ when $\beta\in (0,\beta_*)$.

Eternal solutions to a singular diffusion equation with critical gradient absorption