Partial differential equations of Sobolev-Galpern type

A mixed initial and boundary value problem is considered for a partial differential equation of the form Mut(x, t)+Lu(x, £)=0, where M and L are elliptic differential operators of orders 2 m and 21, respectively, with m ^ I. The existence and uniqueness of a strong solution of this equation in Hι0(G) is proved by semigroup methods. We are concerned here with a mixed initial boundary value problem for the equation (1) Mut + Lu = 0 in which M and L are elliptic differential operators. Equations of this type have been studied using various methods in [2, 3, 4, 6, 7,10,11, 13,14,15,17,18]. We will make use of the ZΛestimates and related results on elliptic operators to obtain a generalized solution to this problem similar to that obtained for the parabolic equation ut + Lu — 0 as in [7]. Let G be a bounded open domain in Rn whose boundary dG is an (n — l)-dimensional manifold with G lying on one side of dG. By Hk(G) = Hk we mean the Hubert space (of equivalence classes) of functions in U(G) whose distributional derivatives through order k be-long to L2(G) with the inner product and norm given, respectively, by (/, 9)k- ΣJ\°fD*gdx: \\ a \\ ̂