### 2006-04

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

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).