Classical solutions of quasilinear parabolic systems on two-dimensional domains
- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 35K40 35K45 35K57
- Partial differential equations, quasilinear parabolic systems, nonsmooth domains, mixed boundary conditions, discontinuous coefficients, local classical solutions, reaction-diffusion systems
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 > 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.
- NoDEA Nonlinear Differential Equations Appl., 13 (2006) pp. 287-310.