Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $u_t (t, x) + Au(t, x) = f (t,x,u, . . . ,∇^m u), (t, x)\in (0, T ) \times \Omega$, 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 $u, . . . ,∇^m u$ 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.