Discontinuous limit semigroups for the singular Zhang equation and its hydrodynamic version (Q5954468)

The article is dedicated to the memory of Anatolii S. Kalashnikov and deals with the properties of the solutions of two semilinear parabolic equations, the singular Zhang equation and its hydrodynamic version, \(u_t = u_{xx} + \ln|u_x|\) and \(v_t = v_{xx} + v_x/v.\) The main aim of the present article is to prove existence, nonexistence and some regularity results for the above-mentioned equations. The investigation is essentially based on the maximum principle and detailed study of families of special solutions. The authors use the concept of the maximal solutions defined by a monotone regular approximation and show that for some classes of initial data, the solutions become instantly entirely singular: \(v(t) \equiv 0\) and \(u(t)\equiv -\infty\) for any arbitrarily small \(t > 0\). The authors establish that for both equations the solutions cannot satisfy the initial conditions in any weak sense. The corresponding semigroups are not continuous at \(t = 0\).