Local regularity of weak solutions of semilinear parabolic systems with critical growth (Q882624)

The semilinear parabolic system \[ u_t + Au= f(t,x,u,\dots, D^mu),\quad (t,x)\in (0,T)\times \Omega, \] \(\Omega \subset \mathbb R^n,\) is considered. Here \(A\) is an elliptic matrix differential operator of order \(2m\) with smooth coefficients that satisfies uniformly the Legendre-Hadamard ellipticity condition. The function \(f\) is of critical growth \[ | f(t,x,p_0,...p_m)| \leq c\biggl(1+ \sum_{i=0}^m | p_i| ^{\frac{n+4m}{n+2i}}\biggr). \] The solution \(u\in L_2((0,T),H^m_2)\cap L_\infty((0,T),L_2)\) is proved to be locally of class \(u\in L_2((T_0,T),H^{2m}_2)\), \(u_t\in L_2((T_0,T),L_2)\). The proofs use the general interpolation inequality and localisation.