by Berchio, E., Grunau, H.-Ch..

**Series:** 2006-04, Preprints

- MSC:
- 35K55 Nonlinear parabolic equations

**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,\nabla^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 $u,\ldots,\nabla^m u$ 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

**This paper was published in:**

J. Evol. Equ.7, 177-196 (2007).