On the existence of the harmonic variational flow subject to the two-sided conditions

We shall devote ourselves to a study of constructing the weak solutions to parabolic systems of the variational flow type associated with the quadratic functional with the initial and boundary data from a suitable Sobolev space, subject to the two-sided conditions. In order to do this, we shall present an approach made by extending Rothe's method which is useful for solving the Cauchy problem and the initial- boundary-value problems for many sorts of equations.