06-04
Local regularity of weak solutions of semilinear parabolic systems with critical growth
by Berchio, E.; Grunau, H.-Ch.
Preprint series: 06-04, Preprints
The paper is published: J. Evol. Equ.7, 177-196 (2007).
- MSC:
- 35K55 Nonlinear PDE of parabolic type
- 35D10 Regularity of generalized solutions
Abstract: We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $$u_t(t,x) + Au(t,x)= f(t,x,u,\ldots,\abla^m u),\quad (t,x) \in (0,T) \times \Omega,$$ is locally regular. Here, $A$ is an elliptic matrix differential operator of order $2m$. The result is proved by writing the system as a system with linear growth in $u,\ldots,\abla^m u$ but with \'bad\' coefficients and by means of a continuity method, where the time serves as parameter of continuity. We also give a partial generalization of previous work of the second author and von Wahl to Navier boundary conditions.
Keywords: local regularity, controllable growth, critical growth
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