Kaiser H.-Ch., Neidhardt H., Rehberg J.
Classical solutions of quasilinear parabolic systems on two dimensional
domains
(531K, Postscript)
ABSTRACT. Using a classical theorem of Sobolevskii on equations of
parabolic type in a Banach space and recently obtained results
on elliptic operators with discontinuous coefficients including
mixed boundary conditions we prove that quasilinear parabolic
systems in diagonal form admit a local, classical solution in the
space of p-integrable functions, for some p greater than 1, over
a bounded two dimensional space domain. As applications we
have in mind systems of reaction diffusion equations, e.g. van
Roosbroeck's system. The treatment of such equations in a
space of integrable functions enables us to define the normal
component of the flow across any part of the Dirichlet boundary
by Gauss' theorem.