The sobolev equation, i

Abstract

The abstract evolution equation Mu?(t)+Lu(t) = f(t) is considered, where the closed operators M and L are obtained from the pairs of Hilbert spaces {V_m,H} and {V₁,H}, respectively. We assume V_l↪V_m↪H, where A↪B means A is a dense subspace of B and the injection is continuous. Under various hypotheses, the existence and uniqueness of weak solutions (in V_l?)and of strong solutions (in H) of the equation are obtained by semi-group methods.We shall consider weak and strong solutions of the Cauchy problem for the abstract evolution equation Mu?(t)+Lu(t) = f(t) in a Hilbert space. In applications, M and L are elliptic partial differential operators of orders 2m and 2l, respectively, with m?l. The definitions of weak and strong solutions are given in Section 1, and sufficient conditions for the existence and uniqueness of a strong solution are given in Section 2. Sufficient conditions for a unique weak solution to exist are given in Section 3, and Section 4 consists of general remarks on the applications of the above results.These applications to partial differential equations together with the regularity of weak solutions and of strong solutions will be discussed in Part II