On the regularity at shrinking points for the porous media equation
In the article under consideration, the authors study bounded solutions of the porous media equation in a noncylindrical domain \(Q_T\) in space-time. It is assumed that for any \(\tau\), \(t_0 < \tau < T\), the domain \(Q_{\tau}\) is homeomorphic, with a \(C^2\) mapping, to a cylinder \(\Omega\times (t_0,\tau)\), where \(\Omega\) has the classical regularity in parabolic problems, and also it is supposed that \(Q_T\) ''shrinks'' at \(P\), that is \[ \overline Q_T\cap\{t=T\} = P, \] where \(P\) is the point \((x = 0, t = T)\); \(P\) is called the ''shrinking point'' of \(Q_T\). The main aim of the article is to give conditions on the domain which ensure the continuity or discontinuity of the solutions of the porous media equation at \(P\). The authors use a recent general result which allows comparisons with the heat equation. This result allows to compare for each \(t\) \(\|u(t) - C\|\) with \(\|v(t) - C\|\) where \(C\) is the constant boundary value, \(u\) solves the porous media equation and \(v\) solves a heat equation in a radially symmetric domain whose measure for \(t\) is the same as for a given domain \(\Omega(t)\). The authors obtain a sufficient continuity condition (at \(P\)) for \(u\), in general domains and a sufficient discontinuity condition for \(u\) in radial domains and prove that the continuity of \(u\) is not a local property of \(Q_T\) by means of an explicit counterexample.
scientific article; zbMATH DE number 1439280