The Sobolev-type equations of the second order with the relatively dissipative operator pencils

Of concern is the Cauchy problem for the Sobolev-type equation of the second order. We introduce the definition of relatively dissipative operator pencils, generalize the notion of dissipativity and relative dissipativity of operators. The connection with the theory of accretive operators is established. According to the Keldysh ideology, the original problem is reduced to the Cauchy problem for the Sobolev-type equation of the first order and the results for the investigated problem are obtained.