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 ut(t, x) +Au(t, x) = f(t, x, u, . . . ,∇u), (t, x) ∈ (0, T )× Ω, 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, . . . ,∇mu 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. Subject–Classification: 35D10, 35K50.