Final value problems for parabolic differential equations and their well-posedness

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax--Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.


INTRODUCTION
In this article we establish well-posedness of final value problems for a large class of parabolic differential equations. Seemingly, this clarifies a longstanding gap in the comprehension of such problems.
Taking the heat equation as a first example, we address the problem of characterising the functions u(t, x) that, in a C ∞ -smooth bounded open set Ω ⊂ R n with boundary ∂ Ω, fulfil the equations, where ∆ = ∂ 2 x 1 + · · · + ∂ 2 x n denotes the Laplacian, (1) Motivation for doing so could be given by imagining a nuclear power plant, which is hit by a power failure at time t = 0. Once power is regained at time t = T , and a measurement of the reactor temperature u T (x) is obtained, it is of course desirable to calculate backwards in time to provide an answer to the question: were temperatures u(t, x) around some earlier time t 0 < T high enough to cause a meltdown of the fuel rods ?
We provide here a theoretical analysis of such problems and prove that they are well-posed, i.e., they have existence, uniqueness and stability of solutions u ∈ X for given data ( f , g, u T ) ∈ Y , in certain normed spaces X , Y to be specified below. The results were announced without proofs in the short note [CJ18].
Although well-posedness is of decisive importance for the interpretation and accuracy of numerical schemes, which one would use in practice, such a theory has seemingly not been worked out before. Explained roughly, our method is to provide a useful structure on the reachable set for a general class of parabolic differential equations.
1.1. Background. Let us first describe the case f = 0, g = 0. Then the mere heat equation (∂ t − ∆)u = 0 is clearly solved for all t ∈ R by the function u(t, x) = e (T −t)λ v(x), if v(x) is an eigenfunction of the Dirichlet realization −∆ D of the Laplace operator with eigenvalue λ .
In view of this, the homogeneous final value problem (1) would obviously have the above u as a basic solution if, coincidentally, the final data u T (x) were given as the eigenfunction v(x). The theory below includes the set B of such basic solutions u together with its linear hull E = span B and a certain completion E .
It is easy to describe E in terms of the eigenvalues 0 < λ 1 ≤ λ 2 ≤ . . . and the associated L 2 (Ω)-orthonormal basis e 1 , e 2 , . . . of eigenfunctions of −∆ D : corresponding to final data u T in span(e j ), which are the u T having finite expansions u T (x) = ∑ j (u T | e j )e j (x) in L 2 (Ω), the space E consists of solutions u(t, x) being finite sums u(t, x) = ∑ j e (T −t)λ j (u T | e j )e j (x).
However, when summation is extended to all j ∈ N, condition (3) becomes very strong, as it is only satisfied for special u T : Weyl's law for the counting function, cf. [CH53,Ch. 6.4], entails the well-known fact λ j = O( j 2/n ), so a single term in (3) yields |(u T | e j )| ≤ c exp(−T j 2/n ); whence the L 2 -coordinates of such u T decay rapidly for j → ∞.
Condition (3) has been known at least since the 1950s; the work of John [Joh55] and Miranker [Mir61] are the earliest we know. While many authors have avoided an analysis of it, Payne found it scientifically intolerable because u T is likely to be imprecisely measured; cf. his treatise [Pay75] on the variety of methods applied to (1) until the mid 1970s.
More recently e.g. Isakov [Isa98] emphasized the classical observation, found already in [Mir61], that (2) implies a phenomenon of instability. Indeed, the sequence of final data u T,k = e k has constant length 1, yet via (2) it gives the initial states u k (0, x) = e T λ k e k (x) having L 2 -norms e T λ k , which clearly blow up rapidly for k → ∞.
This L 2 -instability cannot be explained away, of course, but it does not rule out that (1) is well-posed. It rather indicates that the L 2 -norm is an insensitive choice for (1).
To obtain similarly well-adapted spaces for (1) with f = 0, g = 0, one could base the analysis on (3). Indeed, along with the above space E of basic solutions, a norm |||u T ||| on the final data u T ∈ span(e j ) can be defined by (3), leading to the norm |||u T ||| = (∑ ∞ j=1 e 2T λ j |(u T | e j )| 2 ) 1 2 on the u T that correspond to solutions u in the completion E . This would give well-posedness of (1) with u ∈ E ; cf. Remark 16.
But the present paper goes much beyond this. For one thing, we have freed the discussion from −∆ D and its specific eigenvalue distribution by using sesqui-linear forms, cf. Lax-Milgram's lemma, which allowed us to extend the proofs to a general class of elliptic operators A.
Secondly we analyse the fully inhomogeneous problem (1) for general f , g in Section 5. In this situation well-posedness is not just a matter of choosing the norm on the data ( f , g, u T ) suitably (as one might think from the above |||u T |||). In fact, prior to this choice, one has to restrict the ( f , g, u T ) to a subspace characterised by certain compatibility conditions. While such conditions are well known in the theory of parabolic boundary problems, they are shown here to have a new and special form for final value problems.
Indeed, the compatibility conditions stem from the unbounded operator u T → u(0), which maps the final data to the corresponding initial state in the presence of the source term f . The fact that this operator is well defined, and that its domain endowed with the graph norm yields the data space, is the leitmotif of this article.
1.2. The Abstract Final Value Problem. Let us outline our analysis given for a Lax-Milgram operator A defined in H from a V -elliptic sesquilinear form a(·, ·) in a Gelfand triple, i.e., in a set-up of three Hilbert spaces V ֒→ H ֒→ V * having norms denoted · , | · | and · * , and where V is the form domain of a.
In this framework, we consider the following general final value problem: given data determine the V -valued vector distributions u(t) on ]0, T [ , that is the u ∈ D ′ (0, T ;V ), fulfilling Classically a wealth of parabolic Cauchy problems with homogeneous boundary conditions have been efficiently treated with the triples (H,V, a) and the D ′ (0, T ;V * ) set-up in (6). For this the reader may consult the work of Lions and Magenes [LM72], Tanabe [Tan79], Temam [Tem84], Amann [Ama95]. Also recently, e.g., Almog, Grebenkov, Helffer, Henry studied variants of the complex Airy operator via such triples [AH15,GHH17,GH16], and our results should at least extend to final value problems for those of their realisations that have non-empty spectrum.
To compare (6) with the analogous Cauchy problem, we recall that whenever u ′ + Au = f is solved under the initial condition u(0) = u 0 ∈ H , for some f ∈ L 2 (0, T ;V * ), there is a unique solution u in the Banach space For (6) it would thus be natural to envisage solutions u in the same space X . This turns out to be true, but only under substantial further conditions on the data ( f , u T ).
To formulate these, we exploit that −A generates an analytic semigroup e −tA in B(H). This is crucial for the entire article, because analytic semigroups always are invertible in the class of closed operators, as we show in Proposition 1. We denote its inverse by e tA , consistent with the case −A generates a group, Its domain is the Hilbert space D(e tA ) = R(e −tA ) that is normed by u = (|u| 2 + |e tA u| 2 ) 1/2 . In Proposition 10 we show that a non-empty spectrum, σ (A) = / 0, yields strict inclusions For t = T these domains play a crucial role in the well-posedness result below, cf. (11), where also the full yield y f of the source term f on the system appears, namely The map f → y f takes values in H , and it is a continuous surjection y f : L 2 (0, T ;V * ) → H .
Theorem 1. For the final value problem (6) to have a solution u in the space X in (7), it is necessary and sufficient that the data ( f , u T ) belong to the subspace Y of L 2 (0, T ;V * ) ⊕ H defined by the condition Moreover, in X the solution u is unique and depends continuously on the data (The full statements are found in Theorem 7 and Theorem 8 below.) Condition (11) is a fundamental novelty for the above class of final value problems, but more generally it also gives an important clarification for parabolic differential equations.
As for its nature, we note that the data ( f , u T ) fulfilling (11) form a Hilbert(-able) space Y embedded into L 2 (0, T ;V * ) ⊕ H , in view of its norm in (12).
Using the above y f , (12) is seen to be the graph norm of ( f , u T ) → e TA (u T − y f ), which in terms of Φ( f , u T ) = u T − y f is the unbounded operator e TA Φ from L 2 (0, T ;V * ) ⊕ H to H . As (11) means that the operator e TA Φ must be defined at ( f , u T ), the space Y is its domain. Thus e TA Φ is a key ingredient in the rigorous treatment of (6).
The role of e TA Φ is easy to elucidate in control theoretic terms: its value e TA Φ( f , u T ) simply equals the particular initial state u(0) that is steered by f to the final state u(T ) = u T at time T ; cf. (13) below.
Because of e −(T −t)A and the integral over [0, T ], (11) involves non-local operators in both space and time as an inconvenient aspect -exacerbated by use of the abstract domain D(e TA ), which for longer lengths T of the time interval gives increasingly stricter conditions; cf. (9).
Anyhow, we propose to regard (11) as a compatibility condition on the data ( f , u T ), and thus we generalise the notion of compatibility.
For comparison we recall that Grubb and Solonnikov [GS90] made a systematic investigation of a large class of initial-boundary problems of parabolic (pseudo-)differential equations and worked out compatibility conditions, which are necessary and sufficient for well-posedness in full scales of anisotropic L 2 -Sobolev spaces. Their conditions are explicit and local at the curved corner ∂ Ω × {0}, except for half-integer values of the smoothness s that were shown to require so-called coincidence, which is expressed in integrals over the product of the two boundaries {0} × Ω and ]0, T [ × ∂ Ω; hence it also is a non-local condition.
However, whilst the conditions of Grubb and Solonnikov [GS90] are decisive for the solution's regularity, condition (11) is crucial for the existence question; cf. the theorem.
Showalter [Sho74] attempted to characterise the possible u T in terms of Yosida approximations for f = 0 and A having half-angle π 4 . As an ingredient, invertibility of analytic semigroups was claimed in [Sho74] for such A, but the proof was flawed as A can have semi-angle π/4 even if A 2 is not accretive; cf. our example in Remark 9.
Theorem 1 is proved largely by comparing with the corresponding problem u ′ + Au = f , u(0) = u 0 . It is well known in functional analysis, cf. (7), that this is well-posed for data f ∈ L 2 (0, T ;V * ), u 0 ∈ H , with solutions u ∈ X . However, as shown below by adaptation of a classical argument, u is also in this set-up necessarily given by Duhamel's principle, or the variation of constants formula, for the analytic semigroup e −tA in V * , For t = T this yields a bijective correspondence u(0) ↔ u(T ) between the initial and terminal states (in particular backwards uniqueness of the solutions in the large class X )-but this relies crucially on the previously mentioned invertibility of e −tA ; cf. (8).
As a consequence of (13) one finds the necessity of (11), as the difference Φ( f , u T ) = u T − y f in (11) must equal the vector e −TA u(0), which obviously belongs to D(e TA ).
Moreover, (13) yields that u(T ) in a natural way consists of two parts, that differ radically even when A has nice properties: First, e −tA u(0) solves the semi-homogeneous problem with f = 0, and for u(0) = 0 there is the precise property in non-selfadjoint dynamics that the "height" function h(t) is strictly convex, This is shown in Proposition 4 when A belongs to the broad class of hyponormal operators, studied by Janas [Jan94], or in case A 2 is accretive; then h(t) is also strictly decreasing with The stiffness inherent in strict convexity is supplemented by the fact that u(T ) = e −TA u(0) is confined to a dense, but very small space, as by a well-known property of analytic semigroups, Secondly, for u 0 = 0 the integral in (13) solves the initial value problem, and it has a rather different nature since its final value y f in (10) is surjective y f : L 2 (0, T ;V * ) → H , hence can be anywhere in H , regardless of the Lax-Milgram operator A in our set-up. This we show in Proposition 6 using a kind of control-theoretic argument in case A is self-adjoint with compact inverse; and for general A by means of the Closed Range Theorem, cf. Proposition 5.
For the reachable set of the equation u ′ + Au = f , or rather the possible final data u T , they will be a sum of an arbitrary vector y f in H and a term e −TA u(0) of great stiffness (cf. (15)). Thus u T can be prescribed in the affine space y f + D(e TA ). As any y f = 0 will push the dense set D(e TA ) ⊂ H in some arbitrary direction, u(T ) can be expected anywhere in H (unless y f ∈ D(e TA ) is known a priori). Consequently neither u(T ) ∈ D(e TA ) nor (15) can be expected to hold for y f = 0, not even if its norm |y f | is much smaller than |e −TA u(0)|.
As for final state measurements in real life applications, we would like to prevent a misunderstanding by noting that it is only under the peculiar circumstance that y f = 0 is known a priori to be an exact identity that (15) would be a valid expectation on u(T ).
Indeed, even if f is so small that it is (quantitatively) insignificant for the time development of the system governed by u ′ + Au = f , so that f = 0 is a valid dynamical approximation, the (qualitative) mathematical expectation that u(T ) should fulfill (15) cannot be justified from such an approximation; cf. the above.
In view of this fundamental difference between the problems that are truly and merely approximately homogeneous, it seems that proper understanding of final value problems is facilitated by treating inhomogeneous problems from the very beginning.
1.3. The Inhomogeneous Heat Problem. For (1) with general data ( f , g, u T ) the above is applied with A = −∆ D , that is the Dirichlet realisation of the Laplacian. The results are analogous, but less simple to state and more demanding to obtain.
First of all, even though it is a linear problem, the compatibility condition (11) destroys the old trick of reducing to boundary data g = 0, for when w ∈ H 1 fulfils w = g = 0 on the curved boundary S = ]0, T [ ×∂ Ω, then w lacks the regularity needed to test (11) on the data (f , 0,ũ T ) of the reduced problem; cf. (127) ff.
Secondly, it is, therefore, non-trivial to clarify that every g = 0 does give rise to an extra term z g , in the sense that (11) is replaced by the compatibility condition Thirdly, due to the low reqularity, it requires technical diligence to show that z g , despite the singularity of ∆e (T −s)∆ D at s = T , has the structure of a single convergent improper Bochner integral, namely The reader is referred to Section 5 for the choice of the Poisson operator K 0 and for an account of the results on the fully inhomogeneous problem in (1), especially Theorem 10 and Corollary 3, which we sum up here: Theorem 2. For given data f ∈ L 2 (0, T ; H −1 (Ω)), g ∈ H 1/2 (S), u T ∈ L 2 (Ω) the final value problem (1) is solved by a function u in X 1 = L 2 (0, T ; H 1 (Ω)) C([0, T ]; L 2 (Ω)) H 1 (0, T ; H −1 (Ω)), if and only if the data in terms of (10) and (17) satisfy the compatibility condition (16). In the affirmative case, u is uniquely determined in X 1 and has the representation, with all terms in X 1 , The unique solution u in X 1 depends continuously on the data ( f , g, u T ) in the Hilbert space Y 1 , when these are given the norms in (130) and (158) below, respectively.

Contents.
Our presentation is aimed at describing methods and consequences in a concise way, readable for a broad audience within evolution problems. Therefore we have preferred a simple set-up, leaving many examples and extensions to future work, cf. Section 6. Notation is given in Section 2 together with the set-up for Lax-Milgram operators and semigroup theory. Some facts on forward evolution problems are recalled in Section 3, followed by our analysis of abstract final value problems in Section 4. The heat equation and its final and boundary value problems are treated in Section 5. Section 6 concludes with remarks on the method's applicability and notes on the literature of the problem.

PRELIMINARIES
In the sequel specific constants will appear as C j , j ∈ N, whereas constants denoted by c may vary from place to place. 1 S denotes the characteristic function of the set S.
Throughout V and H denote two separable Hilbert spaces, such that V is algebraically, topologically and densely contained in H . Then there is a similar inclusion into the anti-dual V * , i.e. the space of conjugated linear functionals on V , (V, H,V * ) is also known as a Gelfand triple. Denoting the norms by · , | · | and · * , respectively, there are constants such that for The inner product on H is denoted by (· | ·); and the sesquilinear scalar product on V * × V by ·, · V * ,V or ·, · , it fulfils | u, v | ≤ u * v . The second inclusion in (19) means that for u ∈ H , On an open set Ω ⊂ R n , n ≥ 1, the space C ∞ 0 (Ω) consists of the infinitely differentiable functions having compact support in Ω; it is given the usual L F -topology, cf. [Gru09,Sch66]. The dual space of continuous linear functionals D ′ (Ω) is the distribution space on Ω. We use the standard distribution theory as exposed by Grubb [Gru09] and Hörmander [Hör85].
More generally, the space of B-valued vector distributions is denoted by D ′ (Ω; B); it consists of the continuous linear maps Λ : The Sobolev space H 1 (0, T ; B) consists of the u ∈ D ′ (0, T ; B) for which both u, u ′ belong to L 2 (0, T ; B); it is normed by T 0 ( u 2 + u ′ 2 ) dt) 1/2 . More generally W 1,1 (0, T ; B) is defined by replacing L 2 by L 1 .

Lax-Milgram Operators.
Our main tool will be the Lax-Milgram operator associated to an elliptic sesquilinear form, cf. the set-up in [Gru09,Sect. 12.4]. For the reader's sake we review this, also to establish a few additional points from the proofs in [Gru09].
We let a(·, ·) be a bounded, V -elliptic sesquilinear form on V , i.e., certain C 3 ,C 4 > 0 fulfil for Obviously, the adjoint sesquilinear form a * (u, v) = a(v, u) inherits these properties (with the same C 3 , C 4 ), and so does the "real part", a Re (u, v) = 1 2 (a(u, v) + a * (u, v)). Since a Re (u, u) ≥ 0, the form a Re is an inner product on V , inducing the equivalent norm We recall that s(u, v) = (Su | v) V gives a bijective correspondence between bounded sesquilinear forms s(·, ·) on V and bounded operators S ∈ B(V ), which is isometric since S equals the operator norm of the sesquilinear form |s| = sup |s(u, v)| u = 1 = v . So the given form a induces an A 0 ∈ B(V ) given by and the adjoint form a * similarly induces an operator A * 0 ∈ B(V ), which is seen at once to be the adjoint of A 0 in the sense that ( By Riesz's representation theorem, there exists a bijective isometry J ∈ B(V,V * ) such that for every v * = Jṽ one has Jṽ, v = (ṽ | v) V for all v ∈ V . Therefore A := J • A 0 is an operator in B(V,V * ), for which (24) gives Similarly Clearly A and A ′ are bijections, as composites of such. Hence they give rise to a Hilbert space structure on V * with the inner product inducing the norm |||w||| * = a Re (A −1 w, A −1 w) 1/2 = |||A −1 w||| on V * , equivalent to w * . The Lax-Milgram operator A is defined by restriction of A to an operator in H , i.e., The reader may consult [Gru09, Sect. 12.4] for elementary proofs of the following: A is closed in H , with D(A) dense in H as well as in V ; in H also A ′ has these properties, and it equals the adjoint of A in H ; i.e., A ′ | A ′−1 (H) = A * . As A is closed, D(A) is a Hilbert space with the graph norm v 2 D(A) = |v| 2 + |Av| 2 , and D(A) ֒→ V is bounded due to (22). Geometrically, σ (A) and ν(A) are contained in the sector of z ∈ C given by Both the closed operator A in H and its extension A ∈ B(V,V * ) are used throughout. (For simplicity, they were both denoted by A in the introduction, though.) 2.2. The Self-Adjoint Case. As is well known, if A is selfadjoint, i.e. A * = A (or a * = a), and has compact inverse, then H has an orthonormal basis of eigenvectors of A, which can be scaled to orthonormal bases of V and V * . This is recalled, because our results can be given a more explicit form in this case, e.g. for −∆ in (1).
The properties that A is selfadjoint, closed, and densely defined with dense range in H carry over to A −1 , e.g. [Gru09, Thm. 12.7], so when A −1 in addition is compact in H (e.g., if V ֒→ H is compact), then the spectral theorem for compact selfadjoint operators states that H has an orthonormal basis (e j ) consisting of eigenvectors of A −1 , where the eigenvalues µ j of A −1 by the positivity can be ordered such that The orthonormal basis (e j ) also consists of eigenvectors of A with eigenvalues λ j = 1/µ j .
νA has a bounded inverse for ν = λ j , as ν −1 / ∈ σ (A −1 ). As a Re = a here, V is now renormed by |||v||| 2 = a(v, v). However, if moreover V is considered with a(u, v) as inner product, then A : V → V * is the Riesz isometry; and one has Moreover, the sequence (e j / λ j ) j∈N is an orthonormal basis for V , and Proof. The e j / λ j are orthonormal in V since a(e j , e k ) = A e j , e k = λ j (e j | e k ), cf. (25). They also yield a basis for V since similarly whence the rightmost side converges in V . This means that v = ∑ ∞ j=1 λ j (v | e j )e j / λ j is an orthogonal expansion in V , whence |||v||| 2 has the stated expression.
Fact 2. For every w ∈ V * the expansion w = ∑ ∞ j=1 w, e j e j converges in V * . Moreover, the sequence ( λ j e j ) j∈N is an orthonormal basis of V * and |||w||| 2 so the rightmost side converges in V * , and the expression for |||w||| 2 * results.

Semigroups.
Assuming that the reader is familiar with the theory of semigroups e tA , we review a few needed facts in a setting with a general complex Banach space B. The books of Pazy [Paz83], Tanabe [Tan79] and Yosida [Yos80] may serve as general references. The generator is Ax = lim t→0 + 1 t (e tA x − x), with domain D(A) consisting of the x ∈ B for which the limit exists. A is a densely defined, closed linear operator in B that for certain ω ≥ 0 and M ≥ 1 satisfies The corresponding semigroup of operators is written e tA , it belongs to B(B) with Its basic properties are that e tA e sA = e (s+t)A for s,t ≥ 0, e 0A = I, lim t→0 + e tA x = x for x ∈ B, and the first of these gives at once the range inclusions The following well-known theorem gives a criterion for A to generate an analytic semigroup that is uniformly bounded, i.e., has ω = 0. It summarises the most relevant parts of Theorems 1.7.7 and 2.5.2 in [Paz83], and it involves sectors of the form Theorem 3. If θ ∈ ]0, π 2 [ and M > 0 are such that the resolvent set ρ(A) ⊇ Σ and then A generates an analytic semigroup e zA for | arg z| < θ , for which e zA is bounded for Furthermore, if e tA is analytic, u ′ = Au, u(0) = u 0 is uniquely solved by u(t) = e tA u 0 for every u 0 ∈ B.
Proof. Let e z 0 A u 0 = 0 hold for some u 0 ∈ B, z 0 ∈ S. The analyticity of e zA in S carries over to the map f : z → e zA u 0 , and to By the properties of analytic semigroups (cf. [Paz83, Lem. 2.4.2]) and of u 0 , so that g v ≡ 0 holds on B(z 0 , r) and consequently on S by unique analytic extension.
since e tA is a strongly continuous semigroup. Altogether Z(e z 0 A ) = {0} is proved.
Remark 1. We have only been able to track a claim of the injectivity in Proposition 1 in case z > 0, θ ≤ π/4 and B is a Hilbert space; cf. Showalter's paper [Sho74]. But his proof is flawed, as A 2 is non-accretive for some A with θ ≤ π/4, cf. the counter-example in Remark 9 below.
Remark 2. Injectivity also follows directly when A is defined on a Hilbert space H having an orthonormal basis (e n ) n∈N such that Ae j = λ j e j : Clearly e tA e j = e tλ j e j as both sides satisfy so that v ⊥ e j for all j, and thus v ∈ span(e n ) ⊥ = H ⊥ = {0}. Hence e tA is invertible for such A.
We have chosen to use the symbol e −tA to denote the inverse of the analytic semigroup e tA generated by A, consistent with the case in which e tA does form a group in B(B), i.e., This notation is convenient for our purposes (with some diligence). For simplicity we observe the following when B = H is a Hilbert space and t > 0: clearly e −tA maps D(e −tA ) = R(e tA ) bijectively onto H , and it is an unbounded closed operator in H . As (e tA ) * = e tA * also is analytic, so that Z(e tA * ) = {0} by Proposition 1, we have D(e −tA ) = H , i.e., the domain is dense in H .
A partial group phenomenon and other algebraic properties are collected here: and there is a descending chain of domain inclusions where the right-hand side is a restriction of e −tA e sA . Finally (33) yields (44). .
is undoubtedly known. So let us recall briefly how to prove it strongly measurable, i.e., to find a sequence of simple functions converging pointwise to E(s) f (s) for a.e. s ∈ [0,t]; cf. [Yos80]. Now f can be so approximated by a sequence ( f n ), and E can by its continuity [0,t[ → B(B) also be approximated pointwise for s < t by E n defined on each subinterval [t( j − 1)2 −n ,t j2 −n [ , j = 1, . . ., 2 n , as the value of E at the left end point. Then E f = lim n E n f n on [0,t] a.e. Therefore e (t−·)A f ∈ L 1 (0,t; B) follows directly from (32), Moreover, η, e (t−·)A f is seen to be in L 1 (0,t) by majorizing with e (t−s)A f (s) B η B * , for strong measurability implies weak measurability; cf. Section IV.5 appendix in [RS80]. The main concern is to obtain a Leibniz rule for the derivative: For . Hence a multiplication of the two expansions gives the right-hand side of (47) to the first order in h. The Leibniz rule is more generally valid in the vector distribution sense: Proof. It suffices to cover the case ω = 0, for the other cases then follow by applying the formula to the semigroup e −ωt e tA generated by For arbitrary φ ∈ C ∞ 0 ( ]0, T [ ), we find using the Bochner inequality that

FUNCTIONAL ANALYSIS OF INITIAL VALUE PROBLEMS
Having set the scene in Section 2.1 by recalling elliptic Lax-Milgram operators A in Gelfand triples (V, H,V * ), we now discuss solutions of the classical initial value problem By definition of vector distributions, the above equation means that for every scalar test function First we recall the fundamental theorem for vector functions from [Tem84, Lem. III.1.1]. Further below, it will be crucial for obtaining a solution formula for (53).
Lemma 1. For a Banach space B and u, g ∈ L 1 (a, b; B) the following are equivalent: (i) u is a.e. equal to a primitive function of g, i.e. for some vector ξ ∈ B Remark 4. Lemma 1 is proved in [Tem84], except for the estimate (55): the continuous function u(t) B attains its minimum at some t 0 ∈ [a, b], so applying the Bochner inequality in (i) and the Mean Value Theorem, This yields (55), hence the Sobolev embedding Secondly we recall the Leibniz The well-known generalization below was proved in real vector spaces in [Tem84, Lem. III.1.2] for u = v. We briefly extend this to the general complex case, which we mainly use to obtain that ∂ t |u| 2 = 2 Re u ′ , u , though also u = v will be needed.

Furthermore, u and v have continuous representatives on
Taking We omit a proof of this theorem, as it is a special case of a more general result of Lions and Magenes [LM72, Sect. 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 [LM72], 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 * ).
Alternatively one can use Theorem III.1.1 in Temam's book [Tem84], where proof is given using Lemma 1 to reduce to the scalar differential equation ∂ t u, η + a(u, η) = f , η in D ′ (0, T ), for η ∈ V , which is treated by Faedo-Galerkin approximation and basic functional analysis. His proof extends straightforwardly, from a specific triple (H,V, a) for the Navier-Stokes equations, to the general set-up in Section 2.1, also when A * = A.
However, either way, we need the finer theory described in the next two subsections.
3.2. Well-Posedness. We now substantiate that the unique solution from Theorem 4 depends continuously on the data, so that (53) is well-posed in the sense of Hadamard. First we note that the solution in Theorem 4 is an element of the space X in (60), which is a Banach space when normed, as done throughout, by To clarify a redundancy in this choice, we need a Sobolev inequality for vector functions.

Lemma 3. There is an inclusion L
Proof. If u belongs to the intersection, the continuity follows from Lemma 2, where the formula gives ∂ t |u| 2 = 2 Re u ′ , u . By Lemma 1, integration of both sides entails which by use of the Mean Value Theorem as in Remark 4 leads to the estimate.
Remark 5. In our solution set X in (60) one can safely omit the space C([0, T ]; H), according to Lemma 3. Likewise sup |u| can be removed from · X , as one just obtains an equivalent norm (similarly for the term T 0 u(t) 2 * dt in (7)). Thus X is more precisely a Hilbertable space; we omit this detail in the sequel for the sake of simplicity. However, we shall keep X as stated in order to emphasize the properties of the solutions.
The next result on stability is well known among experts, and while it may be derived from the abstract proofs in [LM72], 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 the data That is, the solution operator Proof. Clearly u ∈ L 2 (0, T ;V ) while the functions u ′ , f and A u belong to L 2 (0, T ;V * ), so as an identity of integrable functions, Hence Lemma 2 and the V -ellipticity gives Using again that |u(t)| 2 and ∂ t |u(t)| 2 are in L 1 (0, T ), taking B = C in Lemma 1 yields For the first two contributions to the X -norm this gives Since u solves (53) it is clear that which upon substitution of (69) altogether shows (64).
3.3. The First Order Solution Formula. We now supplement the well-posedness by a direct proof of the variation of constants formula, which requires that the extended Lax-Milgram operator A generates an analytic semigroup in V * . This is known, cf. [Tan79], but lacking a concise proof in the literature, we begin by analysing A in H : Proof. To apply Theorem 3, we let λ = 0 be given in the sector Σ for some angle θ satisfying 0 < θ < arccot(C 3 /C 4 ). Then it is clear that δ = − sgn(Im λ )θ or δ = 0 gives In case δ ∈ {±θ } a multiplication of the inequalities (28) by − sin δ yields In addition C θ := C 4 cos θ −C 3 sin θ > 0, because cot θ > C 3 C −1 4 . So for u ∈ D(A), Re(e iδ (a(u, u) + λ (u | u))) ≥ Re(e iδ a(u, u)) = cos δ Re a(u, u) − sinδ Im a(u, u) This V -ellipticity holds also if δ = 0, cf. (71), so e iδ (A + λ I) is in any case bijective; and so is −A − λ I. To bound −(A + λ I) −1 , we see from (73) that for u ∈ D(A), This implies (35) for −A. Since A * is the Lax-Milgram operator associated to the elliptic form a * , the above also entails the statement for −A * . For A it follows at once from (73) that Re e iδ (A + λ )u, u ≥ C θ u 2 for u ∈ V . Hence R(A + λ I) is closed in V * , and it is also dense since R(A + λ I) ⊃ R(A + λ I) = H by the above; i.e., A + λ I is surjective. Mimicking (74), we get for u = 0, w = 1, both in V , This yields injectivity of A + λ I and the resolvent estimate. A ′ is covered through a * .
We denote by e −tA the semigroup generated by −A on V * , to distinguish it from e −tA on H . Analogously for e −tA ′ ∈ B(V * ). As A ⊂ A implies that (A + λ I) −1 | H = (A + λ I) −1 , and since A and A have the same sector Σ by Lemma 4, the well-known Laplace transformation formula, cf. [Paz83, Thm. 1.7.7], yields the corresponding fact, say e −tA | H = e −tA for the semigroups: Lemma 5. For all x ∈ H one has e −tA x = e −tA x as well as e −tA ′ x = e −tA * x.
Using Lemmas 4 and 5, the announced formula results as an addendum to Theorem 4: Theorem 5. The unique solution u in X provided by Theorem 4 satisfies that where each of the three terms belongs to X .
Proof. Once (76) has been shown, Theorem 4 applies in particular to cases with f = 0, yielding that u(t) and hence e −tA u 0 belongs to X . For general data ( f , u 0 ) this means that the last term containing f necessarily is a member of X too.
To derive formula (76) in the present general context, one should note that all terms in the equation ∂ t u + A u = f belong to the space L 2 (0, T ;V * ). Therefore the operator e −(T −t)A applies to both sides as an integration factor, yielding Now e −(T −t)A u(t) belongs to L 1 (0, T ;V * ), cf. the argument prior to (46). For its derivative in D ′ (0, T ;V * ) the Leibniz rule in Proposition 3 gives, as As both terms on the right-hand side are in L 2 (0, T ;V * ), the implication (ii) =⇒ (i) in Lemma 1 gives that Proof. It is seen from Theorem 5 that u(t) = e −tA u 0 always is in L 2 (0, T ;V ), as a member of X . By taking scalar products with (T − ·)u on both sides of the differential equation, one obtains in L 1 (0, T ) the identity Taking real parts here, applying Lemma 2 to u and integrating partially on [t, T ], one obtains By reorganising this, a crude estimate yields the result at once for C 5 = C 2 C 1 + 2TC 3 .
3.4. Non-Selfadjoint Dynamics. It is classical that e −tA u 0 in (76) is a term that decays exponentially for t → ∞ if A is self-adjoint and has compact inverse on H . This follows from the eigenfunction expansions, cf. the formulas in the introduction and Section 2.2, which imply for the 'height' function However, it is a much more precise dynamical property that h(t) is a strictly convex function for u 0 = 0 (we refer to [NP06] for a lucid account of convex functions). Strict convexity is established below for wide classes of non-self-adjoint A, namely if A is hyponormal or such that A 2 is accretive.
Moreover, it seems to be a novelty that the injectivity of e −tA provided by Proposition 1 implies the strict convexity. For simplicity we first explain this for the square h(t) 2 .
Indeed, differentiating twice for t > 0 one finds for u = e −tA u 0 , In case A 2 is accretive, that is when m(A 2 ) ≥ 0, we may keep only the last term in (83) to get that (h 2 ) ′′ (t) ≥ 2|Ae −tA u 0 | 2 , which for u 0 = 0 implies (h 2 ) ′′ > 0 as both A and e −tA are injective; cf.
(34) and Proposition 1. Hence h 2 is strictly convex for t > 0 if m(A 2 ) ≥ 0. Another case is when A is hyponormal. For an unbounded operator A this means that Cf. the work of Janas [Jan94]. Note that if both A, A * are hyponormal, then A is normal. This is a quite general class, but it fits most naturally into the present discussion: For hyponormal A we have R(e −tA ) ⊂ D(A) ⊂ D(A * ), which shows that A * e −tA u 0 is defined. Using this and hyponormality once more in (83), we get Now (h 2 ) ′′ > 0 follows for u 0 = 0 from injectivity of e −tA and of A + A * ; the latter holds since 2a Re is V -elliptic. So h 2 is also strictly convex for hyponormal A. Also on the closed half-line with t ≥ 0 there is a result on non-selfadjoint dynamics. Here we return to h(t) itself and normalise, at no cost, to |u 0 | = 1 to get cleaner statements:

Proposition 4. Let A denote a V -elliptic Lax-Milgram operator, defined from a triple (H,V, a), such that A is hyponormal, as above, or such that A 2 is accretive, and let u be the solution from
Theorem 4 for f = 0 and |u 0 | = 1. Then h(t) = |u(t)| is strictly decreasing and strictly convex for t ≥ 0 and differentiable from the right at t = 0 with Proof. By the convexity shown above, (h 2 ) ′ is increasing. Since m(A) > 0 holds by the Vellipticity, h 2 is strictly decreasing (and so is h) for t > 0 as These properties give that , 1[ now extends by continuity to t = 0. So does strict convexity of h, using twice that the slope function is increasing.
By convexity h ′ is increasing for t > 0, so lim t→0 + h ′ (t) = inf h ′ ≥ −∞. For each 0 < s < 1 the continuity of h yields |e −tA u 0 | ≥ s|u 0 | = s for all sufficiently small t ≥ 0. By the above formulas for h ′ and (h 2 ) ′ we have h ′ (t) = − Re(Ae −tA u 0 | e −tA u 0 )/|e −tA u 0 |, so the Mean Value Theorem gives for some τ ∈ ]0,t[ , Hence h(0) > h(t) for all t > 0. Moreover, the limit of h ′ (τ) was shown above to exist for  Remark 8. In Proposition 4 we note that when A 2 is accretive, i.e., m(A 2 ) ≥ 0, then A is necessarily sectorial with half-angle π/4; that is ν(A) ⊂ z ∈ C | arg(z)| ≤ π/4 . This may be seen as in [Sho74,Lem. 3], where reduction to bounded operators was made in order to invoke the operator monotonicity of the square root.
Remark 9. We take the opportunity to point out an error in ([Sho74, ], Lemma 3), where it incorrectly was claimed that having half-angle π/4 also is sufficient for m(A 2 ) ≥ 0. A counterexample is available already for A in B(H) (if dim H ≥ 2), as A = X + iY for self-adjoint X , Y ∈ B(H): here m(A) ≥ 0 if and only if X ≥ 0, and we can arrange that A has half-angle π/4, that is Here we may take Y = δ X + λ 1 U , where δ > 0 is small enough and U is a partial isometry that interchanges two eigenvectors v 1 , v 2 of X with eigenvalues λ 2 > λ 1 > 0, U = 0 on H ⊖ span(v 1 , v 2 ). In fact, , which by the positivity of X and Young's inequality is implied by 1 , that is if 0 < δ ≤ 1 − λ 1 /λ 2 . Now, m(A 2 ) ≥ 0 if and only if |X v| 2 ≥ |Y v| 2 for all v in H , but this will always be violated, as one can see from |Y v| 2 = δ 2 |X v| 2 + λ 2 1 |U v| 2 + 2δ λ 1 Re(X v|U v) by inserting v = v 1 , for the last term drops out as v 1 ⊥ v 2 = U v 1 , so that actually |Y v 1 | 2 = (δ 2 + 1)λ 2 1 > |X v 1 | 2 . Thus A = λ 0 0 4λ + iλ δ 1 1 4δ is a counter-example in C 2 for any λ > 0, 0 < δ ≤ 1/2. Remark 10. It is perhaps useful to emphasize the benefit from joining the two methods. Within semigroup theory the "mild solution" given in (76) is the only possible solution to (53); but as our class of solutions is larger, the extension of the old uniqueness argument in Theorem 5 was needed. Existence of a solution is for analytic semigroups classical if f : [0, T ] → H is Hölder continuous, cf. [Paz83, Cor. 4.3.3]. Using functional analysis, this gap to the weaker condition f ∈ L 2 (0, T ;V * ) is bridged by Theorem 5, which states that the mild solution is indeed the solution in the space of vector distributions in Theorem 4; albeit at the expense that the generator A is a V -elliptic Lax-Milgram operator.

ABSTRACT FINAL VALUE PROBLEMS
In this section, we show for a Lax-Milgram operator A that the final value problem is well-posed when the final data belong to an appropriate space, to be identified below. This is obtained via comparison with the initial value problem treated in Section 3.

A Bijection From
This representation of u(T ) is essential in what follows, as it gives a bijective correspondence u(0) ↔ u(T ) between the initial and terminal states, as accounted for below.
First we analyse the integral term above by introducing the yield map f → y f given by Clearly y f is a vector in V * by definition of the integral (and since C([0, T ];V * ) ⊂ L 1 (0, T ;V * )). But actually it is in the smaller space H , for y f = u(T ) holds in H when u is the solution for u 0 = 0 of (53), and then Corollary 1 yields an estimate of sup t∈[0,T ] |u(t)| by the L 2 -norm of f ; cf. (61). In particular, we have Moreover, f → y f is by (93) bounded L 2 (0, T ;V * ) → H , and it has dense range in H containing all x ∈ D(e εA ) for every ε > 0, for if in (92) we insert the piecewise continuous function then the semigroup property gives y f ε = T T −ε e −εA ( 1 ε e εA x) ds = 1 ε T T −ε x ds = x. However, standard operator theory gives the optimal result, that is, surjectivity: Proposition 5. The yield map f → y f is in B(L 2 (0, T ;V * ), H) and it is surjective, R(y f ) = H . Its adjoint in B(H, L 2 (0, T ;V )) is the orbit map given by v → e −(T −·)A * v.
Proof. To determine the adjoint of f → y f , we first calculate for f ∈ L 2 (0, T ; H) so that the integrand in (92) belongs to C([0, T ]; H). For v ∈ H we get, using the Bochner identity twice, The last scalar product makes sense because s → e −(T −s)A * v is in L 2 (0, T ;V ), as seen by applying Corollary 2 to the Lax-Milgram operator A * , and L 2 (0, T ;V ) is the dual space to L 2 (0, T ;V * ); cf. Remark 11 below. Since L 2 (0, T ; H) is dense in L 2 (0, T ;V * ), it follows by closure that the left-and right-hand sides are equal for every f ∈ L 2 (0, T ;V * ) and v ∈ H . Hence v → e −(T −·)A * v is the adjoint of y f . Applying Corollary 2 to A * for t = 0, a change of variables yields for every v ∈ H , This estimate from below of the adjoint is equivalent to closedness of the range of y f , as the range is dense by (94). This follows from the Closed Range Theorem; cf. [Joh00, Thm. 3.1] for a general result on this.
Proof. Fact 1 yields an ortonormal basis (e n ) n∈N so that Ae n = λ n e n , hence any v in H fulfils v = ∑ j α j e j with ∑ j |α j | 2 < ∞. By Fact 2 every f ∈ L 2 (0, T ;V * ) has an expansion converging in V * for t a.e. Since e −(T −s)A e j = e −(T −s)λ j e j , cf. Remark 2, such f fulfill Hence y f = v is equivalent to the validity of T 0 β j (s)e sλ j ds = α j e T λ j for j ∈ N. So if, in terms of some θ j ∈ ]0, 1[ to be determined, we take the coefficients of f (t) as then the condition will be satisfied if and only if k j = α j e T λ j λ j (e T √ λ j − e θ j T √ λ j ) −1 . Moreover, using the equivalent norm ||| · ||| * on V * in Fact 2, Therefore f is in L 2 (0, T ;V * ) whenever T 0 |β j | 2 dt ≤ Cλ j |α j | 2 holds eventually for some C > 0, and here a direct calculation gives So in view of the expression for k j , the quadratic integrability of f follows if the θ j can be chosen so that the above numerator is estimated by We may take J so that λ j > 3 for all j ≥ J, since at most finitely many eigenvalues do not fulfill this. Then θ j := 1 − (λ j − λ j ) −1 belongs to ]0, 1[ , and the above is reduced to Applying the Mean Value Theorem to exp on [− T √ λ j −1 , 0], we obtain the inequality Hence (103) is fulfilled for C = exp(6T )/T 2 .
Remark 12. In the above proof supp β j ⊂ [θ j T, T ], so the given v can be attained by y f by arranging the coefficients β j in each dimension successively as time approaches T , as θ j ր 1 follows in (99) by counting the eigenvalues so that λ j ր ∞. This can even be postponed to any given T 0 < T , for supp β j ⊂ [T 0 , T ] holds whenever θ j T ≥ T 0 , and we may reset to θ j = T 0 /T and adjust the k j accordingly, for the finitely many remaining j. Both themes may be of interest in infinite dimensional control theory.
In order to isolate u(0) in (91), it will of course be decisive that the operator e −TA has an inverse, as was shown for general analytic semigroups in Proposition 1.
For our Lax-Milgram operator A with analytic semigroup e −tA generated by A = −A, it is the symbol e tA that denotes the inverse, consistent with the sign convention in (41). Hence the properties of e tA can be read off from Proposition 2, where (43) gives Moreover, it is decisive for the interpretation of the compatibility conditions in Section 4.2 below to know that the domain inclusions in Proposition 2 are strict. We include a mild sufficient condition along with a characterisation of the domain D(e tA ).
Proposition 7. If H has an orthonormal basis of eigenvectors (e j ) j∈N of A so that the corresponding eigenvalues fulfil Re λ j → ∞ for j → ∞, then the inclusions in (44) are both strict, and D(e tA ) is the completion of span(e j ) j∈N with respect to the graph norm, The domain D(e tA ) equals the subspace S ⊂ H in which the right-hand side is finite.
Proof. If x ∈ S the vector v = ∑ ∞ j=1 e λ j t (x | e j )e j is well defined in H , and with methods from Remark 2 it follows that e −tA v = x; i.e. x ∈ D(e tA ).
Now any x ∈ D(e t ′ A ) is also in D(e tA ) for t < t ′ , since Re λ j > 0 holds in (106) for all j by V -ellipticity. As Re λ j → ∞, we may choose a subsequence so that Re λ j n > n and set Here x ∈ D(e tA ) as it is in S by construction for t ≥ 0; but not in D(e t ′ A ) for t ′ > t as Furthermore, using orthogonality, it follows for any x ∈ D(e tA ) that, for N → ∞, Hence the space D(e tA ) has span(e j ) j∈N as a dense subspace. That is, the completion of the latter with respect to the graph norm identifies with the former.
After this study of the map y f , the injectivity of the operator e −tA and the domain D(e tA ), cf. Propositions 1, 2, 5 and 7, we address the final value problem (90) by solving (91) for the vector u(0). This is done by considering the map This is composed of the bijection e −TA and a translation by the vector y f , hence is bijective from H to the affine space R(e −TA ) + y f . In fact, using (41), inversion gives This may be summed up thus: Theorem 6. For the set of solutions u in X of the differential equation (∂ t + A )u = f with fixed data f ∈ L 2 (0, T ;V * ), the formulas (91) and (111) give a bijective correspondence between the initial states u(0) in H and the terminal states u(T ) in y f + D(e TA ).
In view of the linearity, the affine space y f + D(e TA ) might seem surprising. However, a suitable reinterpretation gives the compatibility condition introduced in the next section.

4.2.
Well-Posedness of the Final Value Problem. Since R(e TA ) ⊂ H , the initial state in (111) can be inserted into formula (76), so any solution u of (90) must satisfy Here one could contract the first term a bit, as e −tA e TA ⊂ e (T −t)A by (105). But we refrain from this because e −tA e TA rather obviously applies to u T − y f if and only if this vector belongs to D(e TA ) -and the following theorem corroborates that this is equivalent to the unique solvability in X of the final value problem (90): Proof. When (90) has a solution u ∈ X , then u T is reachable from the initial state u(0) determined from the bijection in Theorem 6, which gives that u T − y f = e −TA u(0) ∈ D(e TA ). Hence (113) is necessary and (112) follows by insertion, as explained prior to (112). Uniqueness is obvious from the right-hand side of (112).

Theorem 7. Let V be a separable Hilbert space contained algebraically, topologically and densely in H , and let A be the Lax-Milgram operator defined in H from a bounded V -elliptic sesquilinear form a, and having bounded extension
When u T , f fulfill (113), then u 0 = e TA (u T − y f ) defines a vector in H , so Theorem 4 yields a function u ∈ X solving (∂ t + A )u = f and u(0) = u 0 . According to Theorem 6 this u has final state u(T ) = e −TA e TA (u T − y f ) + y f = u T , hence solves (90).
Finally, the fact that the integral in (112) defines a function in X follows at once from Theorem 5, for it states that it equals the solution in X ofũ ′ + Aũ = f ,ũ(0) = 0. Since u ∈ X in (112), also e −tA e TA (u T − y f ) is a function in X .
Remark 13. When ( f , u T ) fulfils (113), then (111) yields that u T − y f = e −TA u(0). Remark 14. Writing condition (113) as u T = e −TA u(0) + y f , cf. Remark 13, this part of Theorem 7 is natural inasmuch as each set of admissible terminal data u T are in effect a sum of the terminal state, e −TA u(0), of the semi-homogeneous initial value problem (53) with f = 0 and of the terminal state y f of the semi-homogeneous problem (53) with u(0) = 0. Moreover, the u T fill at least a dense set in H , as for fixed u(0) this follows from Proposition 5; for fixed f from the density of D(e TA ) seen prior to Proposition 2.
Remark 15. To elucidate the criterion u T − y f ∈ D(e TA ) in formula (113) of Theorem 7, we consider the matrix operator P A = ∂ t +A r T , with r T denoting restriction at t = T , and the "forward" map Φ( f , u T ) = u T − y f , which by (61) and Proposition 5 give bounded operators Then, in terms of the range R(P A ), clearly (90) has a solution if and only if f u T ∈ R(P A ), so the compatibility condition (113) means that R(P A ) = Φ −1 (D(e TA )) = D(e TA Φ).
The paraphrase at the end of Remark 15 is convenient for the choice of a useful norm on the data. Indeed, we now introduce the space of admissible data Y = D(e TA Φ), i.e.
endowed with the graph norm on D(e TA Φ) given by Using the equivalent norm ||| · ||| * from (26) for V * , the above is induced by the inner product This space Y is complete: as Φ in Remark 15 is bounded, the composite map e TA Φ is a closed operator from L 2 (0, T ;V * ) ⊕ H to H , so its domain D(e TA Φ) = Y is complete with respect to the graph norm given in (116). Hence Y is a Hilbert(-able) space-but we shall often just work with the equivalent norm on the Banach space Y obtained by using simply · * on V * . Moreover, the norm in (116) also leads to continuity of the solution operator for (90): Theorem 8. The solution u ∈ X in Theorem 7 depends continuously on the data ( f , u T ) in the Hilbert space Y in (115), or equivalently, for some constant c we have Another equivalent norm on the Hilbert space Y is obtained by omitting the term |u T | 2 .
Proof. This follows from Corollary 1 by inserting u 0 = e TA (u T − y f ) from (111) into (64), for this gives u 2 X ≤ c|e TA (u T − y f )| 2 + c f 2 L 2 (0,T ;V * ) , where one can add |u T | 2 . Conversely the boundedness of y f and e −TA yield that |u T | 2 ≤ c f 2 + c|e TA (u T − y f )| 2 .
Of course, Theorem 7 and Theorem 8 together mean that the final value problem in (90) is well posed in the spaces X and Y .

THE HEAT EQUATION WITH FINAL DATA
To apply the theory in Section 4, we treat the heat equation and its final value problem. In the sequel Ω stands for a smooth, open bounded set in R n , n ≥ 2 as described in [Gru09, App. C].
In particular Ω is locally on one side of its boundary Γ := ∂ Ω.
For such sets we consider the problem of finding the u satisfying Hereby the trace of functions on Γ is written in the operator notation γ 0 u = u| Γ ; similarly we also use γ 0 for traces on S. r T denotes the trace operator at t = T . We shall also use H 1 0 (Ω), which is the subspace obtained by closing C ∞ 0 (Ω) in the Sobolev space H 1 (Ω). Dual to this one has H −1 (Ω), which identifies with the set of restrictions to Ω from H −1 (R n ), endowed with the infimum norm. The reader is referred to Chapter 6 and Remark 9.4 in [Gru09] for the spaces H s (R n ) and the infimum norm.
For the homogeneous problem with g = 0 in (119) we have the solution and data spaces Here, with y f as the usual integral (cf. (125) below), the data norm in (116) amounts to From Theorems 7 and 8 we may now read off the following result, which is a novelty even though the problem is classical: In the affirmative case, such u are uniquely determined in X 0 and fulfil the estimate Remark 16. For A = −∆ γ 0 one has the equivalent norms in Facts 1, 2 and the characterisation of D(e −T ∆ γ 0 ) in Proposition 7. This is a classical consequence of the compact embedding of H 1 0 (Ω) into L 2 (Ω) for bounded sets Ω (e.g. [Gru09, Thm. 8.2]). Thus one obtains for f = 0, g = 0 the situation described in the introduction, where the space of final data, normed by |||u T |||, via Proposition 7 is seen to be D(e −T ∆ γ 0 ) with equivalent norms. As the completed solution space E in the introduction one may take the Banach space E = X 0 , cf. Theorem 9.
5.2. The Inhomogeneous Case. For non-zero data, i.e., when g = 0 on S, cf. (119), one may of course try to reduce to an equivalent homogeneous problem by choosing a function w so that γ 0 w = g on the surface S. Here we recall the classical Lemma 6. γ 0 : H 1 (Q) → H 1/2 (S) is a continuous surjection having a bounded right inverseK 0 , so w =K 0 g maps every g ∈ H 1/2 (S) to w ∈ H 1 (Q) fulfilling γ 0 w = g and Lacking a reference with details, we note that the lemma is well known for sets like Ω, hence for smooth open bounded sets Ω 1 ⊂ R n+1 with operators γ 0,Ω 1 andK 0,Ω 1 ; cf. Theorem B.1.9 in [Hör85] or Theorem 9.5 in [Gru09] for the flat case. In particular, one can stretch Q to ] − 2T, 2T [ ×Ω and attach rounded ends in a smooth way to obtain a set Ω 1 ⊂ ] − 3T, 3T [ ×Ω equal to Q for 0 < t < T . Here H 1 (Q) = r Q H 1 (Ω 1 ) is a classical result, when the latter space of restrictions to Q has the infimum norm. While H s (∂ Ω 1 ) is defined using local coordinates in a standard way, cf. formula (8.10) in [Gru09], the Sobolev space H s (S) on the surface S can be defined as the set of restrictions r S H s (∂ Ω 1 ). When r Sg = g, thenK 0 g = r QK0,Ω 1g defines the desired operatorK 0 , as γ 0,Ω 1 acts as γ 0 in Q.
Remark 17. The norm in H s (S) can be chosen so that this is a Hilbert space; cf. formula (8.10) in [Gru09]. However, Sobolev spaces on smooth surfaces is a vast subject, requiring so-called distribution densities as explained in [Hör85, Sect. 6.3]. We refer the reader to [Gru09, Sect. 8.2] for a short introduction to this subject; as there, we prefer a more intuitive approach (exploiting the surface measure on Ω 1 ) but skip details. A systematic exposition of this framework can be found in [JHS15,Sect. 4], albeit in a general L p -setting with mixed-norms leading to anisotropic Triebel-Lizorkin spaces F s, a p,q (S) on the curved boundary, which in general are the correct boundary data spaces for parabolic problems with different integrability properties in space and time, as noted in [JS08]; cf. the discussion of the heat equation in [JHS15,Sect. 6.5] and the more detailed account in [MH13,Ch. 7].
However, when splitting the solution of (119) as u = v + w for w as in Lemma 6, then v should satisfy (119) with data (f , 0,ũ T ), At first glance one might therefore think that w is inconsequential for the compatibility condition (125), forũ T − yf there equals the usual term u T − y f minus r T w − y ∂ t w−∆w , where the latter seemingly belongs to D(e −T ∆ γ 0 ) as the pair (∂ t w − ∆w, r T w) could seem to be a vector in the range of the operator P −∆ in Remark 15. But obviously this is not the case, because the function w is outside the domain X 0 of P −∆ . Indeed, w ∈ L 2 (0, T ; H 1 (Ω)) and has γ 0 w = g ≡ 0 in the non-homogeneous case, whence w / ∈ L 2 (0, T ; H 1 0 (Ω)). So one might think it would be necessary to discuss homogeneous problems with larger solution spacesX 0 than X 0 .
We propose to circumvent these difficulties by applying Lemma 6 to the corresponding linear initial value problem instead, since in the present spaces of low regularity there is no compatibility condition needed for this: More precisely, we shall analogously to Section 4 obtain a bijection u(0) ↔ u(T ) between initial and final states by establishing a solution formula as in Theorem 5. (For general background material on (128) the reader could consult Section III.6 in [ABHN11], and for the fine theory including compatibility conditions we refer to [GS90].) Analogously to Theorem 4 and Corollary 1, we depart from well-posedness of (128). This is well known per se, but we need to briefly review the explanation in order to account later for the decisive existence of an improper integral showing up when g = 0 in (119).
As H 1 , H −1 are not dual on Ω, the redundancy in Remark 5 does not extend to the term sup [0,T ] u L 2 above.
As a crucial addendum, we may apply Theorem 5 directly to the function v constructed during the above proof and then substitute v = u − w to derive that This formula for the u solving the inhomogeneous final value problem applies especially for t = T , but we shall keep t in [0, T ] to deduce a formula for its solution.
Our strategy in the following will be to simplify the contributions from w, and ultimately to reintroduce the boundary data g instead of w. To do so, we apply the Leibniz rule in Proposition 3 to our function w in H 1 (0,t; L 2 (Ω)) and get As the first inconvenience, ∆ γ 0 does not commute with the semigroup, since w as an element of H 1 \ H 1 0 belongs to neither the domain of the realization −∆ γ 0 , nor to that of A . Secondly, the right-hand side is only integrable on [0,t − ε] for ε > 0, as the last term has a singularity at s = t ; cf. Theorem 3. As a remedy, we may use the improper Bochner integral Lemma 7. For every w ∈ H 1 (Q) the limit (138) exists in L 2 (Ω) and Proof. As e t∆ γ 0 is uniformly bounded according to Theorem 3 and w ∈ C([0, T ], L 2 (Ω)) was seen in the above proof, bilinearity gives that in L 2 (Ω), Moreover, integration of both sides in (137) gives, cf. Lemma 1, The left-hand side converges by (140), and by dominated convergence the rightmost term does so for ε → 0 + (through an arbitrary sequence), so also t−ε 0 ∆ γ 0 e (t−s)∆ γ 0 w(s) ds converges in L 2 (Ω) as claimed. Then (139) is the resulting identity among the limits.
Identity (139) from the lemma applies directly in the solution formula (136), and because terms with ∂ s w cancel, one obtains We shall reduce the difference of the last two integrals in order to reintroduce the boundary data g instead of w. First we use that ∆ = A A −1 ∆ on H 1 (Ω) and write both terms as improper integrals, Here Q = I − A −1 ∆ is a well-known projection from the fine elliptic theory of the problem In fact, if this is treated via the matrix operator −∆ γ 0 , which has an inverse in row form ( −A −1 K 0 ) that applies to the data f g , the basic composites appear in the two operator identities on H 1 (Ω) and H −1 (Ω) ⊕ H 1/2 (Γ) respectively, Thus we get from the first formula that Q = I − A −1 ∆ = K 0 γ 0 on H 1 (Ω). However, before we implement this, we emphasize that the simplicity of the formulas (145) and (146) relies on a specific choice of K 0 explained in the following: As A = ∆ H 1 0 holds in the distribution sense, P := A −1 ∆ clearly fulfils P 2 = P, is bounded H 1 → H 1 0 and equals I on H 1 0 , so P is the projection onto H 1 0 (Ω) along its null space, which evidently is the closed subspace of harmonic H 1 -functions, namely Therefore H 1 is a direct sum, We also let Q = I − P denote the projection on Z(−∆) along H 1 0 (Ω), as from the context it can be distinguished from the time cylinder (also denoted by Q).
Since γ 0 : H 1 (Ω) → H 1/2 (Γ) is surjective with H 1 0 as the null-space, it has an inverse K 0 on the complement Z(−∆), which by the open mapping principle is bounded Hence K 0 : H 1/2 (Γ) → H 1 (Ω) is a bounded right-inverse, i.e. γ 0 K 0 = I H 1/2 (Γ) . The rest of (146) follows at once. Moreover, since γ 0 P = 0, which by definition of Q and P gives (145). (K 0 is known as a Poisson operator; these are amply discussed within the pseudo-differential boundary operator calculus in [Gru96].) Using this set-up we obtain: Proposition 9. If u denotes the unique solution to the initial boundary value problem (128) provided by Proposition 8, then u fulfils the identity where the improper integral converges in L 2 (Ω) for every t ∈ [0, T ].
For t = T the second term in (151) gives back y f = T 0 e −(T −s)A f (s) ds from Section 4. However, the full influence on u(T ) from the boundary data g is collected in the third term as That the map g → z g is well defined is clear by taking t = T in Proposition 9; this is a non-trivial result. The map is linear by the calculus of limits. In case f = 0, u 0 = 0 it is seen from (151) that z g = u(T ), so obviously z g L 2 (Ω) ≤ sup t u(t) L 2 (Ω) , which in turn is estimated by c g H 1/2 (S) using Proposition 8. This proves Lemma 8. The linear operator g → z g is bounded H 1/2 (S) → L 2 (Ω).
Finally, from Proposition 9, we conclude for an arbitrary solution in X 1 of the heat equation Therefore we also here have a bijection u(0) ↔ u(T ), for the above breaks down to application of the bijection e T ∆ γ 0 , cf. Proposition 1, and a translation in L 2 (Ω) by the fixed vector y f − z g . We are now ready to obtain unique solvability of the inhomogeneous final value problem (119). Our result for this is similar to the abstract Theorem 7 (as is its proof), except for the important clarification that the boundary data g do appear in the compatibility condition, but only via the term z g : Theorem 10. For given data f ∈ L 2 (0, T ; H −1 (Ω)), g ∈ H 1/2 (S), u T ∈ L 2 (Ω) the final value problem (119) is solved by a function u ∈ X 1 , whereby if and only if the data in terms of (92) and (152) satisfy the compatibility condition In the affirmative case, u is uniquely determined in X 1 and has the representation where the three terms all belong to X 1 as functions of t .
Proof. Given a solution u ∈ X 1 , the bijective correspondence yields u T = e T ∆ γ 0 u(0) + y f − z g , so that (155) necessarily holds. Inserting its inversion u(0) = e −T ∆ γ 0 (u T − y f + z g ) into the solution formula from Proposition 9 yields (156); thence uniqueness of u. If (155) does hold, u 0 = e −T ∆ γ 0 (u T − y f + z g ) is a vector in L 2 (Ω), so the initial value problem with data ( f , g, u 0 ) can be solved by means of Proposition 8. Then one obtains a function u ∈ X 1 that also solves the final value problem (119), since in particular u(T ) = u T is satisfied, cf. the bijection (153) and the definition of u 0 .
The final regularity statement follows from the fact that X 1 also is the solution space for the initial value problem in Proposition 8. Indeed, even the improper integral is a solution in X 1 to (128) with data ( f , g, u 0 ) = (0, g, 0), according to Proposition 9; cf. the proof of Lemma 8. Similarly the integral containing f solves an initial value problem with data ( f , 0, 0), hence is in X 1 . In addition, the first term in (156) solves (128) for data (0, 0, e −T ∆ γ 0 (u T − y f + z g )).
We let Y 1 stand for the set of admissible data. Within L 2 (0, T ; H −1 (Ω)) ⊕ H 1/2 (Γ) ⊕ L 2 (Ω) it is the subspace given, via the map Φ 1 ( f , g, u T ) = u T − y f + z g , as Correspondingly we endow Y 1 with the graph norm of the operator e −T ∆ γ 0 Φ 1 , that is, of the composite map ( f , g, u T ) → e −T ∆ γ 0 (u T − y f + z g ). Again, e −T ∆ D Φ 1 ( f , g, u T ) equals the initial state u(0) steered by f , g to the final state u(T ) = u T , as is evident for t = 0 in (156).
Recalling that A = −∆ : H 1 0 (Ω) → H −1 (Ω), the above-mentioned graph norm is given by Here the last term is written with explicit integrals to emphasize the complexity of the fully inhomogeneous boundary and final value problem (119). Completeness of Y 1 follows from continuity of Φ 1 , cf. Lemma 8 concerning z g . Indeed, its composition to the left with the closed operator e −T ∆ γ 0 in L 2 (Ω) (cf. Proposition 2) is also closed. Hence its domain D(e −T ∆ γ 0 Φ 1 ) = Y 1 is complete with respect to the graph norm in (158). As this norm is induced by an inner product when the norm of H −1 (Ω) is taken as ||| · ||| * from (26), and when H 1/2 (Q) is normed as in Remark 17, Y 1 is a Hilbert(-able) space.
Analogously to the proof of Theorem 8, continuity of ( f , g, u T ) → u is now seen at once by inserting the expression u 0 = e −T ∆ γ 0 (u T − y f + z g ) from (153) into the estimate in Proposition 8. Thus we obtain: Within semigroup theory on a Banach space B, the above means that the ranges R(e tA ) should form a strictly descending chain of inclusions in the sense that, for t ′ > t > 0, Non-strictness is here characterised by the rather special spectral properties of A in (iv): Theorem 11. For a C 0 -semigroup e tA with e tA ≤ Me ωt the following are equivalent: (i) e tA is injective and R(e t ′ A ) = R(e tA ) holds for some t,t ′ with t ′ > t ≥ 0.
(ii) e tA is injective with range R(e tA ) = B for every t ≥ 0.
(iii) The semigroup is embedded into a C 0 -group G(t) satisfying G(t) ≤ Me ω|t| ; (iv) The spectrum σ (A) is contained in the strip in C where −ω ≤ Re λ ≤ ω and Proof. Given (i) for t > 0, then R(e (t+δ )A ) = R(e tA ) holds for all δ ∈ [0,t ′ − t] in view of the inclusions (33); and to every x ∈ B some y satisfies e tA e δ A y = e tA x, which by injectivity gives x = e δ A y, so that e δ A is surjective for such δ . Hence e tA = (e (t/N)A ) N is a bijection on B with bounded inverse, i.e., 0 ∈ ρ(e tA ). If (i) holds for t = 0, clearly 0 ∈ ρ(e t ′ A ). In both cases (ii) holds because 0 ∈ ρ(e sA ) must necessarily hold for s > 0 according to [Paz83, Thm. 1.6.5], which also states that (iii) holds. (The proof there uses [Paz83, Lem. 1.6.4], that can be invoked directly from (ii) since the inverse of e tA is bounded by the Closed Graph Theorem.) Conversely (iii) yields R(e tA ) = R(G(t)) = B and injectivity for all t ≥ 0, so (ii) and hence (i) holds. That (iv) =⇒ (iii) is part of the content of [Paz83, Thm. 1.6.3], which also states that (iii) implies (iv) for real λ , but the full statement in (iv) is then obtained from [Paz83, Rem. 1.5.4].
This result is essentially known, but nonetheless given as a theorem, as it clarifies how widely the present paper applies. Indeed, for V -elliptic Lax-Milgram operators A, the semigroups are uniformly bounded, so ω = 0; thus the strip in (iv) is the imaginary axis i R, but this is contained in ρ(A) by Lemma 4. So except in the pathological case σ (A) = / 0, (iv) will always be violated, as will (i) and (ii). However, since in (i) and (ii) the operator e −tA is injective by Proposition 1, the strict inclusions in (159) Therefore, for elliptic Lax-Milgram operators A with non-empty spectrum, the compatibility condition (113) is without redundancy, and it gets effectively stronger on longer time intervals. Previously, these properties were verified only in a special case in Proposition 7.
Example 1. It is illuminating to consider the final value problem on R n , for α ∈ C \ R, At first glance this might seem to be a minor variation on the heat problem in Section 5, in fact just a zero-order perturbation; and notably a change to Ω = R n . However, interestingly it cannot be treated within the present framework: in a paper fundamental to analysis of the Stark effect, Herbst [Her79] proved for the operator h(α) = −∆ +αx 1 I with Im α = 0 that the minimal realisationh(α) also is maximal in L 2 (R n ) with empty spectrum, σ (h(α)) = / 0.
Thereforeh(α) is not sectorial, as ν(h(α)) ⊂ ν(h(α)) shows that (28) does not hold, so existence and uniqueness for the forward problem cannot be derived from Theorem 4. The fact proved in [Her79] that e − ith(α)/α is a contraction semigroup, which for α = i applies to e −th(i) that pertains to (162), entails via the Hille-Yosida theorem the estimate in Theorem 11 (iv) for −h(i), but only for Re λ > 0. Since Re λ < 0 is not covered, it is despite the empty spectrum of A = −h(i) = ∆ − i x 1 not clear whether (159) holds with strict inclusions. Thus it seems open which properties final value problem (162) for the Herbst operator h(α) can be shown to have.
Remark 18. Recently Grebenkov, Helffer and Henry [GHH17] studied the complex Airy operator A = −∆+i x 1 in dimension n = 1. They considered realizations defined on R + by Dirichlet, Neumann and Robin conditions using the Lax-Milgram lemma, so results on boundary homogenous final value problems for − d 2 dx 2 + i x should be straightforward to write down, as in Section 5.1. The study was extended to dimension n = 2, under the name of the Bloch-Torrey operator, by Grebenkov and Helffer in [GH16], where bounded and unbounded domains with C ∞ boundary was treated; in cases with non-empty spectrum there should be easy consequences for the associated final value problems. The realisations induced by a transmission condition at an interface, which was the main theme in [GHH17,GH16], are defined from a recent extension of the Lax-Milgram lemma due to Almog and Helffer [AH15], so in this case the properties of the corresponding final value problems are as yet unclear.
Remark 19. We expect that extension of the theory to certain systems of parabolic equations with prescribed boundary and final value data should be possible. A useful framework for the discussion of this type of problems could be the pseudo-differential boundary operator calculus, with matrix-formed operators acting in Sobolev spaces of sections of vector bundles, as described in Section 4.1 of [Gru96]. At least the present discussion should carry over to this kind of problems when the realisations called (P + G) T there are variational, i.e., when they are Lax-Milgram operators for certain triples (H,V, a); this property is analysed in great depth in Section 1.7 of [Gru96], to which we refer the interested reader. It is conceivable that the variational property is unnecessary, and might be avoided using the pseudo-differential boundary operator calculus, but this seems to require an addition to the theory of parabolic systems covered by the calculus in the form of a result on backward uniqueness. 6.2. Notes. Classical considerations were collected by Liebermann [Lie05] for second order parabolic differential operators (cf. also Evans [Eva10]), with references back to the fundamental L 2 -theory including boundary points of Ladyshenskaya, Solonnikov and Uraltseva [LSU68]. A fundamental framework of functional analysis for parabolic Cauchy problems was developed by Lions and Magenes [LM72]. 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 [GS90], who obtained the necessary and sufficient compatibility conditions on the data, including coincidence for half-integer values of the smoothness; cf. also [Gru96,Thm. 4.1.2]. This study was carried over to the corresponding anisotropic L p -Sobolev spaces by Grubb [Gru95]. A further extension to different integrability properties in time and space was taken up in a systematic study of anisotropic mixed-norm Triebel-Lizorkin spaces on a time cylinder and its flat and curved boundaries by Munch Hansen, the second author and Sickel [JHS15]. Compatibility conditions were addressed for the heat equation in mixed-norm Triebel-Lizorkin spaces in [JHS15, Sect. 6.5] and [MH13,Ch. 7]. In particular the latter showed that, except for coincidence at half integer smoothness, the recursive formulation of the compatibility conditions in [GS90] is equivalent to the requirement that the data belong to the null space of a certain matrix-formed operator at the curved corner {0} × ∂ Ω. Recent semigroup and Laplace transformation methods were exposed in [ABHN11]. Denk and Kaip [DK13] treated parabolic multi-order systems via the Newton polygon and obtained L p -L q regularity results using R -boundedness.
To our knowledge, the literature contains no previous account for pairs of spaces X and Y in which final value problems for parabolic differential equations are well posed.
An early contribution on final value problems for the heat equation was given in 1955 by John [Joh55], who dealt with numerical aspects. In 1961, the idea of reducing the data space to obtain well-posedness was adopted by Miranker [Mir61] for the homogeneous heat equation on R, and he showed that in the space of L 2 -functions having compactly supported Fourier transform there is a bijection between the initial and terminal states.
In addition to the injectivity of analytic semigroups in Proposition 1, it is known that u(0) is uniquely determined from u(T ) even for t -dependent sesquilinear forms a(t; v, w). This was shown by Lions and Malgrange [LM60] with an involved argument. It would take us too far to quote the large amount of work on the backward uniqueness in more loosely connected situations, often adopting the log-convexity method (if |u(t)| ≤ |u(T )| t/T |u(0)| 1−t/T then u(T ) = 0 implies u(t) = 0 for all t > 0, hence u(0) = 0 by continuity) attributed to Krein, Agmon and Nirenberg. Instead we refer the reader to [Kuk07,HD11,DE18] and the references therein.
The method of quasi-reversibility for final value problems was introduced systematically in 1967 by Lattès and Lions [LL67]. The idea is to perturb the equation u ′ + Au = 0 by adding, e.g., −ε 2 A 2 to obtain a well-posed problem and to derive for its solution u ε that u ε (x, T ) approaches u T for ε → 0, circumventing analysis of well-posedness of the original final value problem. They assumed f = 0 for a V -elliptic self-adjoint A.
Showalter [Sho74] addressed questions that were partly similar to ours. He proposed to perturb instead by εA∂ t under the condition that A is m-accretive with semiangle θ ≤ π/4 on a Hilbert space for f = 0. He claimed uniqueness of solutions, and existence if and only if the final data via the Yosida approximations of −A allow approximation of the initial state. Showalter also identified injectivity of operators in analytic semigroups as an important tool. However, his reduction had certain shortcomings; cf. Remark 1. In comparison we obtain the full wellposedness for general f = 0 and V -elliptic operators of semiangle θ = arccot(C 3 C −1 4 ) belonging to the larger interval ]0, π/2[ . An extensive account of the area around 1975, and of the many previous contributions using a variety of techniques, was provided by Payne [Pay75]. A more recent exposition can be found in Chapters 2 and 3 in Isakov's book [Isa98], and for methods for inverse problems in general the reader may consult Kirsch [Kir96].
In the closely related area of exact and null controllability of parabolic problems, the inequality in Corollary 2 is a little weaker than the observability inequality for the full subdomain O = Ω. In this context the role of observability was reviewed by Fernandez-Cara and Guerrero [Fer06], emphasising Carleman estimates as a powerful tool in the area. A treatise on Carleman estimates in the parabolic context was given by Koch and Tataru [KT09].