Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

1. Introduction. In this paper we establish the null controllability of linear and semilinear parabolic equations with dynamic boundary conditions of surface diffusion type. The prototype of such problems is ∂ t y − ∆y = 1 ω v(t, x) in (0, T ) × Ω, ∂ t y Γ − ∆ Γ y Γ + ∂ ν y = 0 on (0, T ) × Γ, y Γ (t, x) = y| Γ (t, x) on (0, T ) × Γ, (y, y Γ )| t=0 = (y 0 , y 0,Γ ) in Ω × Γ. (1.1) For all given T > 0, ω Ω and initial data y 0 ∈ L 2 (Ω) in the bulk and y 0,Γ ∈ L 2 (Γ) on the boundary, we want to find a control v ∈ L 2 ((0, T ) × ω) such that the solution satisfies y(T, ·) = 0 in Ω. Here Ω ⊂ R N is a bounded domain with smooth boundary Γ = ∂Ω, N ≥ 2, and the control region ω is an arbitrary nonempty open subset which is strictly contained in Ω (i.e., ω ⊂ Ω). Further, y| Γ denotes the trace of a function y : Ω → R, ν is the outer unit normal field, ∂ ν y := (ν · ∇y)| Γ is the normal derivative at Γ, and ∆ Γ designates the Laplace-Beltrami operator on Γ. In our main results we study more general problems involving time-depending potentials, forcing and semilinear terms The term ∂ t y Γ −∆ Γ y Γ models the tangential diffusive flux on the boundary which is coupled to the diffusion equation in the bulk by the normal derivative ∂ ν y. In (1.1) we treat this problem as a coupled system of dynamic equations for y and y Γ , with side condition y| Γ = y Γ . Sometimes this type of boundary conditions is called of Wentzell type. Dynamic surface and interface processes have attracted a lot of attention in recent years in the mathematical and applied literature, see [3,5,7,8,14,15,16,17,18,26,30,34]. In particular, for the mathematical theory of surface diffusion boundary conditions we refer to [5,8,14,26,30,34]. In Section 2 we discuss the L 2 -based solution theory for (1.1) and its generalizations as needed in the context of null controllability. Here we look at existence, uniqueness and regularity of strong, mild and distributional solutions. Since we deal with timedepending potential terms in L ∞ , we include proofs.
We state our main result ensuring the null controllability of (1.1), see Theorem 4.2. We emphasize that the initial data y 0 and y 0,Γ on Ω and Γ need not be related.
Null controllability results of this type are known for Dirichlet and for inhomogeneous or nonlinear Neumann boundary conditions (also called Robin or Fourier boundary conditions), see e.g. [4,6,9,10,13,19,24] and the survey article [11]. Our Theorem 4.2 actually treats a more general version of the control problem (1.1) with nonautonomous potential terms and inhomogeneities in the bulk and on the boundary. We also show null controllability for a semilinear variant of (1.1) including globally Lipschitz nonlinearities both in the bulk and on the boundary, see Theorem 4.4. Observe that one could drop the assumption that ω ⊂ Ω by simply extending the control v by 0. We will not do this since one would lose information in this way. As in the case of Dirichlet or Neumann boundary conditions, in Theorem 4.5 we obtain null controllability with controls acting on a part Γ 0 of the boundary by means of an auxiliary domain control problem on an enlarged spatial domain, see [4,6,13,19]. However, in our case one needs more regularity for the solution of the problem to implement this strategy. We partly solve this problem by means of local L p -regularity, but we do not obtain the anticipated regularity of the solution and the boundary control v near the boundary of the support of v. In view of the results in [6,9,10,11], we expect that one can extend the results to nonlinearities with slightly superlinear growth and containing gradient terms, as in [12], using more involved regularity theory for (1.1). In this work, we have considered the heat equation with constant diffusion coefficients, but as in the case of static boundary conditions presumably our results also hold for general elliptic second order operators, with diffusion coefficients even depending on time. Recently, many results are obtained even in the case where diffusion coefficients degenerate at the boundary, see [1] and the references therein.
We are not aware of results on null controllability for parabolic problems with dynamical boundary conditions. Optimal control problem and approximate controllability for such equations were treated in [2] and [18] in the case of global controls; i.e., ω = Ω. In [22], approximate boundary controllability of a one-dimensional heat equation with dynamical boundary conditions was studied by completely different methods.
Theorem 1.1 relies on a Carleman estimate for the inhomogeneous dual problem corresponding to (1.1), which is proved in Lemma 3.2. Roughly speaking, this estimate bounds a weighted L 2 -norm of the solution ϕ to the dual problem by a weighted L 2 -norm of the inhomogeneities and of the restriction ϕ| ω . These weights tend to zero exponentially as t → 0 and t → T . The proof of our Carleman estimate follows the known strategy of the Dirichlet case, see [11], but the dynamic boundary condition leads to various new boundary terms. Some of these enter in the final estimate, a few cancel, and others can be controlled using the smoothing effect of the surface diffusion in (1.1). Since we have to use the smoothing effect, we cannot treat the corresponding equations without surface diffusion, see Remark 3.3 for more details.
With the Carleman estimate at hand, standard arguments (see Proposition 4.1) yield the observability estimate for the solution ϕ of the dual homogeneous backward problem in Ω × Γ.
One calls this property the final state observability since the observation on ω controls the state at the final time. By duality, the observability estimate for ϕ then yields the null controllability of (1.1) as stated in Theorem 1.1, see Theorem 4.2. We refer to [33] for a discussion of various controllability and observability concepts.
The proof of the null controllability of the semilinear equation relies on Schauder's fixed point theorem. To set up this fixed point argument, we construct a control with minimal weighted L 2 -norm for the inhomogenous linear system involving nonautonomous potential terms. This optimization problem is solved in Proposition 4.3. In its proof we adopt the methods of Imanuvilov's seminal paper [19] to the case of dynamic boundary conditions. This paper is organized as follows. In Section 2 we introduce the functional analytic setting and state basic results for (versions of) (1.1) concerning well-posedness in the framework of strong, mild and distributional solutions. The Carleman estimate is proved in Section 3 and the null controllability results are obtained in Section 4.

2.
The initial-boundary value problem. Let T > 0 and a bounded domain Ω ⊂ R N , N ≥ 2, with smooth boundary Γ = ∂Ω and outer unit normal field ν on Γ be given. We write where ω Ω is open. In this section we present wellposedness and regularity properties of solutions of the inhomogeneous linear system in Ω T , (2.1) for given coefficients d, δ > 0, a ∈ L ∞ (Ω T ) and b ∈ L ∞ (Γ T ). We include proofs since the potentials depend on time and are just L ∞ and since we also deal with very weak solution concepts.
2.1. Function spaces. The Lebesgue measure on Ω and the surface measure on Γ are denoted by dx and dS, respectively. We consider the real Hilbert spaces (and tacitly their complexifications if necessary) The scalar product on L 2 is given by Further, H k (Ω) are the usual L 2 -based Sobolev spaces over Ω. The spaces H k (Γ) are defined via local coordinates, see e.g. Definition 3.6.1 in [32]. At a few points we will also need the fractional order spaces H s (Ω) and H s (Γ) with noninteger s ≥ 0. For our purposes it suffices to define them as interpolation spaces where (·, ·) s/2,2 denotes the real interpolation functor, see Chapter 1 and Theorem 4.3.1/2 of [32] or Chapter 1 of [25]. As a consequence, we obtain the interpolation inequalities For every s > 1 2 , the trace operator on Γ is continuous and surjective from H s (Ω) to H s−1/2 (Γ) and has a continuous right-inverse E Γ : H s−1/2 (Γ) → H s (Ω), see e.g. Theorem 4.7.1 in [32]. Given s 0 > 1 2 , the right-inverse can be chosen to be independent of s < s 0 . The normal derivative ∂ ν y = (ν · ∇y)| Γ is thus continuous from H s (Ω) to H s−3/2 (Γ) for each s > 3 2 . Finally, for open sets ω ⊂ Ω, we consider L 2 (ω) as a closed subspace of L 2 (Ω) by extending functions on ω by zero to Ω.

2.2.
The Laplace-Beltrami operator. We refer to Chapter 3 of [20] or Sections 2.4 and 5.1 of [31] for more details and proofs. The operator ∆ Γ on Γ is given by in local coordinates g, where G = (g ij ) is the metric tensor corresponding to g and G −1 = (g ij ) denotes its inverse. However, in this paper we will not use this local formula, but rather the surface divergence theorem where ∇ Γ is the surface gradient and ·, · Γ is the Riemannian inner product of tangential vectors on Γ. The Laplace-Beltrami operator with domain H 2 (Γ) is selfadjoint and negative on L 2 (Γ), cf. p. 309 of [31], and it thus generates an analytic C 0 -semigroup on L 2 (Γ). Hence, y L 2 (Γ) + ∆ Γ y L 2 (Γ) defines an equivalent norm on H 2 (Γ). Moreover, y L 2 (Γ) + ∇ Γ y L 2 (Γ) gives an equivalent norm on H 1 (Γ).
2.3. The Laplacian with surface diffusion boundary conditions. On L 2 we consider the linear operator Observe that E 1 = H 1 (0, T ; L 2 ) ∩ L 2 (0, T ; D(A)). Strong solutions of (2.1)-(2.4) will belong to E 1 . We show that A is selfadjoint and negative. The well-posedness and regularity results for the underlying evolution equations rely on this fact, which mainly follows from a result in [27].
2.4. Existence, uniqueness and regularity of solutions. We are concerned with the following classes of solutions of (2.1)-(2.4).
We show below that a strong solution is a mild one and that mild and distributional solutions coincide in our setting. Since our controllability results rely on an observability estimate for a dual problem, we also have to look at the adjoint backward evolution equation in Ω T , (2.9) (ϕ(T, ·), ϕ Γ (T, ·)) = (ϕ T , ϕ T,Γ ) in Ω × Γ, (2.12) for given (ϕ T , ϕ T,Γ ) in H 1 or in L 2 , f ∈ L 2 (Ω T ) and g ∈ L 2 (Γ T ). As in Definition 2.3, a strong solution of (2.9)-(2.12) is a function (ϕ, ϕ Γ ) ∈ E 1 fulfilling (2.9)-(2.12) in L 2 (0, T ; L 2 ), and a mild solution of (2.9)-(2.12) is a function x), one can pass from statements about (2.1)-(2.4) to those about (2.9)-(2.12), and vice versa, by means of the transformation t = T − t. Hence, the following results on strong and mild solutions have straightforward analogues for the adjoint problem which can easily be proved by this transformation. We omit the details, but establish in Proposition 2.5(f) a 'solution formula' for homogenous backward system which is crucial for our main Theorem 4.2.
We start with strong solutions of (2.1)-(2.4). Proposition 2.2 implies that such a solution can only exist if y 0 ∈ H 1 ; i.e., the initial data on Ω and Γ are related by the trace.
Then there exists a unique strong solution Y := (y, y Γ ) ∈ E 1 of (2.1)-(2.4), which is also a mild solution. Given R > 0, there is a constant C = C(R) > 0 such that for all a and b with a ∞ , b ∞ ≤ R and all data we have (2.14) Proof. We set F = (f, g) and Since A is selfadjoint and negative and B(·) is uniformly bounded, Theorem 3.1 of [28] yields the asserted unique solution Y := (y, y Γ ) ∈ E 1 of (2.1)-(2.4) and the estimate (2.14).
so that Y is also a mild solution of (2.1)-(2.4). To obtain the asserted uniformity of the constant in (2.14), we note that Theorem 3.1 of [28] gives bounded linear operators S(t, s) on L 2 depending strongly continuously on 0 ≤ s ≤ t ≤ T such that Taking F = 0 and varying the initial time, (2.15) yields at first for Y 0 ∈ H 1 and then for Y 0 ∈ L 2 by approximation. From Gronwall's inquality we now deduce that S(t, s) ≤ C = C(R) for all 0 ≤ s ≤ t ≤ T . Due to (2.16), the strong solution of (2.1)-(2.4) thus satisfies for all t ∈ [0, T ]. We further writeF = BY + F so that ∂ t Y = AY +F . The estimates (2.14) for a = b = 0 and (2.18) finally yield C = C(R) with We next consider mild and distributional solutions for initial data in L 2 .
Then the following assertions are true.
There are bounded linear operators S(t, s) on L 2 depending strongly continuous on 0 ≤ s ≤ t ≤ T such that to Proposition 2.4 with a = b = 0. Using the self-adjointness of A and integration by parts in time, we thus obtain for all (ϕ, ϕ Γ ) ∈ E 1 with ϕ(T ) = ϕ Γ (T ) = 0. By approximation, this identity also holds for Y 0 ∈ L 2 . Now, let Y ∈ C([0, T ]; L 2 ) be the mild solution of (2.1)-(2.4).
With the notation of the proof of Proposition 2.4 we have for each (ϕ, ϕ Γ ) as above. Another application of (2.21) with Y 0 = 0 and F = BY yields that here the second term on the left-hand side equals To prove that a distributional solution is already the mild one, we show uniqueness of distributional solutions. Let Y, Z ∈ L 2 (0, T ; L 2 ) be such solutions. We then obtain The version of Proposition 2.4 for the backward problem (2.9)-(2.12) says that for every (ψ, . Therefore Y = Z, and uniqueness follows. (e) Let Y be a distributional solution with vanishing end value. Parts (a) and Integrating and using the self-adjointness of A and B(t), we then derive Since every (ψ, ψ Γ ) ∈ H 2 can be represented as (ψ, ψ Γ ) = (ϕ(T ), ϕ Γ (T )) with a function (ϕ, ϕ Γ ) as above and H 2 is dense in L 2 , we conclude that Y (T ) = 0.
We note that for strong solutions of (1.1) with v = 0 one has the dissipiation equality , and not only the estimate (2.20).
3. The Carleman estimate. In this section we prove a Carleman estimate for the backward adjoint linear problem (2.9)-(2.12), which is the key to null controllability in the linear and semilinear case. The weights appearing in the Carleman estimate are the same as in [11] for the case of Dirichlet boundary conditions and in [13] for mixed boundary conditions. They are based on the following auxiliary function η 0 , see Lemma 1.2 in [11] and Lemma 1.1 in [13].
for some constant c > 0.
Given ω Ω, we take λ, m > 1 and η 0 with respect to ω as in Lemma 3.1. Following [11], we then define the weight functions α and ξ by for x ∈ Ω and t ∈ (0, T ). Note that α and ξ are C 2 and strictly positive on (0, T )×Ω and blow up as t → 0 and as t → T . Moreover, the weights are constant on the boundary Γ so that We state the Carleman estimate. In the proof we follow the strategy of Lemma 1.3 of [11]. In our setting several new boundary terms arise from the dynamic boundary condition. To collect and treat them, we have to repeat some steps from [11] in modified form. Define η 0 , α and ξ as above with respect to ω . Then there are constants C > 0 and λ 1 , s 1 ≥ 1 such that for all λ ≥ λ 1 , s ≥ s 1 and (ϕ, ϕ Γ ) ∈ E 1 . Given R > 0, the constant C = C(R) can be chosen independently of all a, b with a ∞ , b ∞ ≤ R.
Proof. Rescaling in time, we restrict to the case d = 1. It can be seen by convolution with mollifiers in space and time that C ∞ ([0, T ] × Ω) is dense in E 1 . Since all terms in the asserted inequality are continuous with respect to the E 1 -norm, it suffices to consider smooth functions ϕ ∈ C ∞ ([0, T ] × Ω). For such functions we write ϕ instead of ϕ Γ . Throughout C denotes a generic constant which does not depend on λ, s, a, b, and ϕ subject to the assumptions of the lemma.
Step 1. Change of variables. Let ϕ ∈ C ∞ ([0, T ]×Ω), λ ≥ λ 1 ≥ 1 and s ≥ s 1 ≥ 1 be given. Define Observe that these functions vanish exponentially at t = 0 and t = T . We determine the problem solved by ψ. We first expand the spatial derivaties of α by the chain rule to bring η 0 into play, but we do not expand ∂ t α. We calculate On Ω T this yields transformed evolution equations Similarly, using (3.1) and (3.2), on Γ T we obtain Extending the corresponding decomposition in [11], we rewrite the equations (3.5) and (3.6) as with the abbreviations Applying · 2 L 2 (Ω T ) resp. · 2 L 2 (Γ T ) to the equations in (3.7) and adding the resulting identities, we obtain

LAHCEN MANIAR, MARTIN MEYRIES AND ROLAND SCHNAUBELT
Step 2. Estimating the mixed terms in (3.8) from below. We often use the following basic pointwise estimates on Ω, Step 2a. We start with the negative term Using integration by parts, (3.3) and (3.1), we further derive For sufficiently large λ 1 , the fact that ∇η 0 = 0 on Ω\ω implies for some C = C(Ω, ω). Integrating by parts in time, we continue with where we employed (3.9) and that ψ(0) = ψ(T ) = 0. This term is absorbed by (3.10) for large λ 1 . Altogether, we have shown using also (3.1). We remark that first and third terms in the last line are the dominant positive terms involving ψ 2 , whereas the second term will lead to a 'control term' on the right hand side of the final estimate.
Step 2b. Integration by parts and (3.2) yield As above, the first summand will lead to a term controlling |∇ψ| 2 . We now apply Young's inequality to (λ 2 ξ 1/2 ψ) ξ 1/2 ∇(|∇η 0 | 2 ) · ∇ψ and sλψξ ∇η 0 · ∇ψ , respectively, and estimate It follows The next summand is given by ∇η 0 · ∇|∇ψ| 2 ξ dx dt due to integration by parts and (3.3). In the sum (3.11), the third term is nonnegative, and the second one can be bounded by The last term in the above line can be estimated as in (3.12). We then infer Because ∇ψ vanishes at t = 0 and t = T in view of (3.4), we obtain We summarize the inequalities of this step and invoke again that ∇η 0 = 0 on Ω\ω . Using also (3.1), we arrive at i=1,2,3 increasing λ 1 and s 1 if necessary.
Step 2c. Employing (3.9), we estimate This term will be absorbed by (3.10) for large λ. Integration by parts, (3.3) and (3.9) next imply Integrating by parts with respect to time, we can derive since ψ vanishes at the endpoints and |∂ 2 t α| ≤ Cξ 3 . We conclude from the above inequalities that i=1,2,3 Step 2d. We now consider the boundary terms N 1 and N 2 , employing the surface divergence theorem (2.6) several times. We first compute by means of ψ(0) = ψ(T ) = 0. Moreover, (3.2) yields The next two terms are estimated by where we proceed as in (3.14) and use (3.9), respectively. Finally, the summand cancels with the one from (3.13), and Step 3. The transformed estimate. We collect the final inequalities in Steps 2a-2d. Increasing λ 1 and s 1 if needed to absorb lower order terms, we arrive We combine this estimate with (3.8). The expressions forf andg lead to additional lower order terms which can be absorbed to the left-hand side for large λ 1 and s 1 .
Using also |∇ψ| 2 = |∇ Γ ψ| 2 + |∂ ν ψ| 2 and (3.1), we deduce We denote the four latter boundary integrals on the right-hand side of (3.15) by I 1 , . . . , I 4 . Young's inequality allows to estimate I 1 by Increasing s 1 if necessary, we can then control (3.16) by the left-hand side of (3.15). The fourth boundary integral I 4 is treated analogously. For the integral I 3 , we have Increasing again s 1 and λ 1 if necessary, (3.17) can be absorbed by the left-hand side of (3.15). For the last integral I 2 , we use the identity (2.6) and that ξ(t, ·) is constant on Γ. We then obtain The second summand in (3.18) can be absorbed by the left-hand side of (3.15) choosing a sufficiently large λ 1 . Altogether, we thus arrive at To put the last summand in (3.19) to the left, we observe that δ∆ Γ ψ = N 2 ψ − sψ∂ t α + ∂ ν ψ. Combined with (3.9), this identity yields for sufficiently large s 1 . We can now choose sufficiently large λ 1 and s 1 so that (3.19) becomes Step 4. Inverting the transformation. The inequality (3.20) allows to replace in (3.21) the summand N 2 ψ 2 L 2 (Γ T ) by the term I times a constant, where we increase λ 1 and s 1 if necessary to absorb the lower order terms in (3.20). Similarly, from ∂ t ψ = N 1 ψ − sλψξ∂ ν η 0 , we deduce that and hence also I can be put on the left hand side of (3.21). In a similar way one handles the corresponding terms on Ω T , see (1.58) and (1.59) in [11]. We thus infer also using ω ⊂ ω. As on p.1409 of [11], one can now absorb the gradient term on the right-hand side by the integral on ω T and the left-hand side. It remains to insert ψ = e −sα ϕ into (3.22). The terms involving derivatives of ψ then lead to various lower order terms which can be controlled by the other terms in (3.22). For summands on Ω T this is done in Step 3 of the proof of Lemma 1.3 in [11]. Since the new terms on the boundary can be treated in the same way, we omit the details. One thus obtains the asserted Carleman estimate for the original function ϕ. Remark 3.3. Up to inequality (3.15), the arguments in the above proof remain valid also for δ = 0. However, in our proof the assumption δ > 0 is essential to put the third and fourth boundary integral I 3 and I 4 on the right-hand side of (3.15) to the left. 4. Null controllability. In this section we apply the Carleman estimate to show null controllability for (1.1) and its generalizations. Throughout we fix T > 0, ω Ω, d, δ > 0, a ∈ L ∞ (Q T ) and b ∈ L ∞ (Γ T ). We assume that s 1 and λ 1 are sufficiently large to apply Lemma 3.2 for this data.
We now establish the null controllability of the linear system, where we allow for inhomogeneities with exponential decay at t = 0 and t = T . To this end, we introduce the weighted L 2 -spaces In Proposition 4.3 we weaken the assumptions on f and g, requiring decay only at t = T .
The proof of null controllability in the semilinear case is based on a fixed point argument involving a continuous operator mapping the initial value to a null control of the inhomogeneous linear problem. To obtain such a operator, we have to single out a special control. As in Theorem 2.1 of [19] we choose a control having a minimal norm. We prefer to state this more technical point in a proposition below, separated from Theorem 4.2.
To that purpose, for given a ∈ L ∞ (Ω T ) and b ∈ L ∞ (Γ T ) we define the backward parabolic operator As in [19] we consider the weights which only blow up at t = T . We introduce the weighted spaces Z Ω := f ∈ L 2 (Ω T ) : e sαξ−3/2 f ∈ L 2 (Ω T ) , endowed with the corresponding scalar products as above. Observe that the weights force decay only at t = T . We further define the functional J by The general strategy of the following proof is the same as in [19], even though we have not been able to obtain the Lagrange multiplier as in (2.5)-(2.6) of [19]. Hence our arguments differ in certain important points from [19], and we thus give all the details. There is a constant C > 0 such that for all Y 0 ∈ L 2 , f ∈Z Ω and g ∈Z Γ we have (4.11) Given R > 0, the constant C = C(R) can be chosen independently of all a, b with a ∞ , b ∞ ≤ R.
Proof. Since M is a closed convex subset of X × L 2 (ω T ) and J 1/2 is an equivalent norm ofX ×L 2 (ω T ), the functional J has a unique minimizer for given Y 0 , f and g if we can show that the set M is nonempty. We will construct a function (Y * , v * ) ∈ M as a weak limit of minimizers of regularized problems.
Therefore, J has a unique minimizer (Y, v) ∈ M on the nonempty subset M . Using the weak convergence and (4.14), we derive the estimate (4.11) for (Y, v) by Step 4. Finally, let f ∈Z Ω and g ∈Z Γ be the given inhomogeneities. Consider the solution set M with respect to f and g. Choose f n ∈Z Ω and g n ∈Z Γ with compact support in (0, T ] such that f n → f inZ Ω and g n → g inZ Γ as n → ∞. Let (Y n , v n ) be the corresponding minimizers of J obtained in Step 3 (where M = M n is defined for f n and g n ). Since these functions satisfy (4.11), we find a subsequence such that (Y nj , v nj ) tends to some (Y * , v * ) weakly inX × L 2 (ω T ) as j → ∞. The limit (Y * , v * ) is a distributional solution of (4.1)-(4.4) with Y * (T, ·) = 0 so that (Y * , v * ) ∈ M . This implies as before that J has a unique minimizer (Y, v) on M . The estimate (4.11) for (Y, v) can be shown as in (4.15).
SinceF ,G are bounded, the estimate (4.11) in Proposition 4.3 shows that Φ 1 maps all Y ∈ L 2 (0, T ; L 2 ) into a ball in L 2 (0, T ; L 2 ). Moreover, Proposition 2.5 says that Φ 1 (Y ) is even a strong solution of (4.20). Proposition 2.4 and (4.11) thus yield where C does not depend on Y . Since E 1 is compactly embedded into L 2 (0, T ; L 2 ) by Proposition 2.2, we conclude that Φ 1 is compact.