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Eternal solutions to a singular diffusion equation with critical gradient absorption

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