\(L^\infty\)-estimates for parabolic systems with VMO-coefficients (Q983437)

The authors consider parabolic systems on \(\mathbb R^n\) in non-divergence form of higher order \[ u_t+\sum_{|\alpha|\leq m}a_\alpha D^\alpha u=0,\qquad u(\cdot,0)=u_0, \] where \(a_\alpha\in \text{VMO}(\mathbb R^n; \mathbb C^{N\times N})\) for \(|\alpha|=m.\) There are obtained a priori estimates for the corresponding autonomous resolvent problem in the \(L^\infty(\mathbb R^n)^N.\) As a consequence it is obtained that the realization of \(A=\sum_{|\alpha|\leq m}a_\alpha D^\alpha\) in \(L^\infty (\mathbb R^n)^N\) is a sectorial operator and generates a (not strongly continuous) semigroup on \(L^\infty(\mathbb R^n)^N.\)