Eternal solutions to a singular diffusion equation with critical gradient absorption

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