Eternal solutions to a singular diffusion equation with critical gradient absorption

Abstract

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-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 $\beta$ ranges in a bounded interval $(0,{\beta }_{\ast }]$ which is in sharp contrast with well-known singular diffusion equations such as ${\partial }_t\varphi -{\Delta }_p\varphi =0$ when $p=2N/(N+1)$ or the porous medium equation ${\partial }_t\varphi -\Delta {\varphi }^m=0$ when $m=(N-2)/N$. Moreover, the profile $f(r;\beta )$ decays to zero as $r\to \infty$ in a faster way for $\beta ={\beta }_{\ast }$ than for $\beta \in (0,{\beta }_{\ast })$ but the algebraic leading order is the same in both cases. In fact, for large $r$, $f(r;{\beta }_{\ast })$ decays as $r^{-p/(2-p)}$ while $f(r;\beta )$ behaves as $(\log r{)}^{2/(2-p)}{r}^{-p/(2-p)}$ when $\beta \in (0,{\beta }_{\ast })$.

References in corpus (3)

• Positivity, decay, and extinction for a singular diffusion equation with gradient absorption
• Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption
• Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Cited by in corpus (5)

• Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction
• Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids
• Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction
• Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption
• Large time behavior for a quasilinear diffusion equation with critical gradient absorption