Matrix quasi-elliptic operators in a"e n

Matrix quasi-elliptic operators in the entire space that belongs to quasi-elliptic operators and includes homogeneous elliptic operators, elliptic, and parabolic operators, is studied. The unique solvability of initial value problems for a wide class of Sobolev-type systems is proved using isomorphism theorems. It is shown that weighted Sobolev spaces can be used in the proof of isomorphism theorems for the Laplacian in the space. The class of homogeneous quasi-elliptic operators is found to include scalar quasi-elliptic operators. The weighted Sobolev spaces of vector functions with the smoothness and power weights defined by vectors is considered and this vector function is found to be inside the space. The solvability of system is proved by constructing approximate solutions to some classes of differential equations.