Solvability and completeness of solutions of parabolic differential- operator equations

We consider an abstract Cauchy problem for parabolic differential-operator equations in Hilbert spaces. Initial boundary value problems for parabolic equations are reduced to the Cauchy problem for a system of parabolic differential equations. It is proved that the solution of an initial boundary value problem for partial parabolic equation can be approximated by linear combinations of elementary solutions. Completeness of elementary solutions is also proved for differential-operator equations in abstract Hilbert spaces. The obtained abstract results are applied to differential equations.