Classification of razor blades to the filtration equation -- the sublinear case.
The author studies non-negative solutions in the sense of \(D'(\mathbb R^n \setminus \{0\} \times (0, + \infty)) \) of the equation \(u_t=\Delta \phi (u).\) Here \(n>2\) and \(\phi\) is bounded between two powers (that are smaller than 1) near the origin. The initial data are identically zero on \(\mathbb R^n \setminus\{0\}.\) A \textit{razor blade} is a solution which is not zero nor a fundamental solution. Depending on the behavior of \(\phi\) near \(u=0\), different situations are found: (A) non-existence of razor blades, (B) existence of razor blades that are for all times not locally integrable near the origin, (C) solutions that satisfy in \(D'(\mathbb R^n \times (0, + \infty))\) the equation \(u_t=\Delta \phi(u) +\mu\) with \(\mu\) a Radon measure supported on \(\{0\} \times (0, \tau)\) (this measure is attained as a trace, and called the axial trace) while having a standing singularity for times greater than \(\tau\). The existence of razor blades with a prescribed axial trace is shown, and that the razor blade is uniquely determined by its axial trace. The author then classifies the behavior in time of all possible (self-similar) razor blade solutions in the case \(\phi(u)=u^m.\)
scientific article; zbMATH DE number 1884216
scientific article