The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition
Volker Pluschke, Frank Weber · Czech digital mathematics library · 1999
The evolution problem which is investigated shows the following special features : (i) The time derivative is multiplied by a coefficient which may vanish in certain time-dependent subdomains. Hence, the differential equation we consider is parabolic-elliptic. (ii) The Lr-function g(\\Delta; \\Delta; ¸), arising in the boundary condition Bu = g(\\Delta; \\Delta; u), is only assumed to be defined and bounded on f ¸ 2 R : j¸j R g. The local weak solvability is proven by means of the Rothe method which is based on a semidiscretization with respect to the time variable, whereby the given problem is approximated by a sequence of linear elliptic problems. In view of (ii), the approximations, obtained by solving these "discretized" problems, have to be estimated in L1 . For that purpose, we use a Moser-technique, where Lp-estimates uniformly approach the desired boundedness statement as p \\Gamma! 1. For the treatment of the degenerate differential equation, this L1 -technique is combined with ...