Partial Differential Equations of Sobolev-galpern Type

A mixed initial and boundary value problem is considered for a partial differential equation of the form Mu t (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 H ι 0 (G) is proved by semigroup methods. We are concerned here with a mixed initial boundary value problem for the equation (1) Mu t + Lu = 0 in which M and L are elliptic differential operators. Equations of this type have been studied using various 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 u t + Lu — 0 as in [7]. Let G be a bounded open domain in R n whose boundary dG is an (n — l)-dimensional manifold with G lying on one side of dG. By H k (G) = H k we mean the Hubert space (of equivalence classes) of functions in U(G) whose distributional derivatives through order k belong to L 2 (G) with the inner product and norm given, respectively, by HI' ΞΞΞ H o k (G) will denote the closure in H k of C~(G), the space of infinitely differentiate functions with compact support in G. The operators are of the form M = Σ {(-l) ιpι D p m pσ (x)D σ : \p\,\σ\^m} and L = Σ {(-iy pι DΠ pσ (x)D σ : \p\,\σ ^ 1} ,