WIAS Preprint No. 765, (2002)

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 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.

Appeared in

  • NoDEA Nonlinear Differential Equations Appl., 13 (2006) pp. 287-310.

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