Existence and uniqueness of very singular solutions for a fast diffusion equation with gradient absorption. (English) Zbl 1264.35137
Summary: Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption
\[
\partial _tu-\Delta_pu+|\nabla u|^q =0\quad \text{ in } (0,\infty)\times \mathbb R^N,
\]
where \(2N/(N+1)<p<2\) and \(p/2<q<p - N/(N+1)\), thereby extending previous results restricted to \(q>1\).
MSC:
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35K67 | Singular parabolic equations |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
| 35B33 | Critical exponents in context of PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 35B07 | Axially symmetric solutions to PDEs |
