Non-Homogeneous Boundary Value Problems and Applications

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

“…For this the reader may consult the work of Lions and Magenes [8], Tanabe [9], Temam [10], Amann [11]. Also recently, e.g., Almog, Grebenkov, Helffer, Henry studied variants of the complex Airy operator via such triples [12][13][14], and our results should at least extend to final value problems for those of their realisations that have non-empty spectrum.…”

“…Clearly the conjunction of u ∈ L 2 (0, T; V) and u ∈ L 2 (0, T; V * ), which appears in [8], is equivalent to the claim in (60) that u belongs to the intersection of L 2 (0, T, V) and H 1 (0, T; V * ).…”

“…The next result on stability is well known among experts, and while it may be derived from the abstract proofs in [8], we shall give a direct proof based on explicit estimates: Corollary 1. The unique solution u of (53), given by Theorem 4, depends continuously as an element of X on…”

“…We omit a proof of this theorem, as it is a special case of a more general result of Lions and Magenes ( [8], Section 3.4.4) on t-dependent forms a(t; u, v). Clearly the conjunction of u ∈ L 2 (0, T; V) and u ∈ L 2 (0, T; V * ), which appears in [8], is equivalent to the claim in (60) that u belongs to the intersection of L 2 (0, T, V) and H 1 (0, T; V * ).…”

“…also Evans [34]), with references back to the fundamental L 2 -theory including boundary points of Ladyshenskaya, Solonnikov and Uraltseva [35]. A fundamental framework of functional analysis for parabolic Cauchy problems was developed by Lions and Magenes [8]. Later a full regularity theory in scales of anisotropic L 2 -Sobolev spaces was worked out for general pseudo-differential parabolic problems by Grubb and Solonnikov [15], who obtained the necessary and sufficient compatibility conditions on the data, including coincidence for half-integer values of the smoothness; cf.…”