On a Cahn-Hilliard type phase field system related to tumor growth

The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.

The physical context is that of a tumor-growth model which has been derived from a general mixture theory [16,12]. We also refer to [4], [15] and [6] for overview articles and to [7] and [5] for the study of related sharp interface models.
We point out that the unknown function u is an order parameter which is close to two values in the regions of nearly pure phases, say u 1 in the tumorous phase and u −1 in the healthy cell phase; the second unknown µ is the related chemical potential, specified by (1.2) as in the case of the viscous Cahn-Hilliard or Cahn-Hilliard equation depending on whether α > 0 or α = 0 (see [3,9,10]); the third unknown σ stands for the nutrient concentration, typically σ 1 in a nutrient-rich extracellular water phase and σ 0 in a nutrient-poor extracellular water phase.
In the case that the parameter α is strictly positive, the problem (1.1)-(1.5) is a generalized phase field model, while it becomes of pure Cahn-Hilliard type in the case that α = 0. On the other hand, the presence of the term αµ t in (1.1) gives, in the case α > 0, a parabolic structure to equation (1.1) with respect to µ. Let us note that the meaning of the coefficient α here differs from the one in (1.2): in (1.1) α is not a viscosity coefficient since it enters in the natural Lyapunov functional of the system, which reads (cf. [13]) However, the fact that the coefficients are taken equal in the system (1.1)-(1.3) is somehow related to the limiting problem obtained by formal asymptotics on α. Indeed, we refer to the forthcoming article [13] for a formal study of the relation between these models and the corresponding sharp interfaces limits.
We remark that the original model deals with functions W and p that are precisely related to each other. Namely, we have p(u) = 2p 0 W(u) if |u| ≤ 1 and p(u) = 0 otherwise (1.6) where p 0 is a positive constant and where W(u) := − u 0 f (s) ds is the classical Cahn-Hilliard double well free-energy density. However, this relation is not used in this paper, whose first aim is proving the well-posedness of the initial-boundary value problem (1.1)- (1.5) in the case of a positive parameter α. In this setting, we can allow W to be even a singular potential (the reader can see the later Remark 2.1).
Actually, we prove the existence of a unique strong solution to the system (1.1)-(1.5) under very general conditions on p and W, as well as we study the long time behavior of the solution; in particular, we can characterize the omega limit set and deduce an interesting property in a special physical case (cf. the later Corollary 2.5). Next, in a more restricted setting for the double-well potential W, we investigate the asymptotic behavior of the problem as the coefficient α tends to zero and find the convergence of subsequences to weak solutions of the limiting problem. Moreover, under a smoothness condition on the initial value u 0 we are able to show uniqueness for the limit problem and consequently also the convergence of the entire family as α 0 (see Theorem 2.6).
Our paper is organized as follows. In Section 2, we state our assumptions and results on the mathematical problem. The forthcoming sections are devoted to the corresponding proofs. In Section 3, we prove the uniqueness of the solution. After presenting a priori estimates in Section 4, we prove the existence of the solution on an arbitrary time interval in Section 5, while we study its large time behavior in Section 6; Section 7 is devoted to the study of the limit of the phase field model (1.1)-(1.5) to the corresponding Cahn-Hilliard problem as α → 0.

Statement of the mathematical problem
In this section, we make precise assumptions and state our results. First of all, we assume Ω to be a bounded connected open set in R 3 (lower-dimensional cases could be considered with minor changes) whose boundary Γ is supposed to be smooth. As in the Introduction, the symbol ∂ ν denotes the (say, outward) normal derivative on Γ. As the first aim of our analysis is to study the well-posedness on any finite time interval, we fix a final time T ∈ (0, +∞) and let Q := Ω × (0, T ) and Σ := Γ × (0, T ). and endow the spaces (2.2) with their standard norms, for which we use a self-explanatory notation like · V . If p ∈ [1, +∞], it will be useful to write · p for the usual norm in L p (Ω). In the sequel, the same symbols are used for powers of the above spaces. It is understood that H ⊂ V * as usual, i.e., in order that u, v = Ω uv for every u ∈ H and v ∈ V , where · , · stands for the duality pairing between V * and V .
As far as the structure of the system is concerned, we are given two constants α and γ and three functions p, β and λ satisfying the conditions listed below α ∈ (0, 1) and γ > 0 (2.3) p : R → R is nonnegative, bounded and Lipschitz continuous (2.4) β : R → [0, +∞] is convex, proper, lower semicontinuous (2.5) λ ∈ C 1 (R) is nonnegative and λ is Lipschitz continuous. (2.6) We define the potential W : R → [0, +∞] and the graph β in R × R by W := β + λ and β := ∂ β (2. 7) and note that β is maximal monotone. In the sequel, we write D( β ) and D(β) for the effective domains of β and β, respectively, and we use the same symbol β for the maximal monotone operators induced on L 2 spaces.
Remark 2.1. Note that lots of potentials W fit our assumptions. Typical examples are the classical double well potential and the logarithmic potential defined by where the decomposition W = β + λ as in (2.7) is written explicitely. More precisely, in (2.9), κ is a positive constant which does or does not provide a double well depending on its value, and the definition of the logarithmic part of W log is extended by continuity at ±1 and by +∞ outside [−1, 1]. Moreover, another possible choice is the following taking the value 0 in [−1, 1] and +∞ elsewhere. Clearly, if β is multi-valued like in the case (2.10), the precise statement of problem (1.1)-(1.5) has to introduce a selection ξ of β(u). We also remark that our assumptions do not include the relationship (1.6) between W and p, and one can wonder whether what we have required is compatible with (1.6). This is the case if β and λ satisfy suitable conditions, in addition. For instance, we can assume the following: D( β ) includes the interval [−1, 1] and β vanishes there; λ is strictly positive in (−1, 1) and λ(±1) = λ (±1) = 0. In such a case, W presents two minima with quadratic behavior at ±1 and the function p given by (1.6) actually satisfies (2.4). We note that this excludes the case of the logarithmic potential, while it includes both (2.8) and (2.10).
As far as the initial data of our problem are concerned, we assume that µ 0 , u 0 , σ 0 ∈ V and β (u 0 ) ∈ L 1 (Ω) (2.11) while the regularity properties which we obtain for the solution are the following (2.13) At this point, the problem we want to investigate consists in looking for a quadruplet (µ, u, σ, ξ) satisfying the above regularity and the following boundary value problem a.e. on Σ (2.17) µ(0) = µ 0 , u(0) = u 0 and σ(0) = σ 0 in Ω .
We note once and for all that adding (2.14) and (2.16) yields Here is our well-posedness result. Once we know that there exists a unique solution on any finite time interval, we can address its long time behavior. Our next result deals with the omega limit of an arbitrary initial datum satisfying (2.11). Even though the possible topologies are several, by recalling (2.12) we choose Φ : as a phase space and set strongly in Φ for some {t n } +∞ . We deduce an interesting consequence in a special case which, however, is significant. Indeed, if σ s = γµ s , (2.23) implies u ω ∈ D( β ) and p(u ω ) = 0 a.e. in Ω. On the other hand, we have u ω ∈ V , whence the conclusion. We also remark that, in the case of the potential (2.10), the inclusion in (2.23) reduces to β(u ω ) µ s . In particular, u ω = −1 if µ s < 0 and u ω = 1 if µ s > 0.
Our final result regards the asymptotic analysis as the viscosity coefficient α tends to zero and the study of the limit problem. We can deal with this by restricting ourselves to a particular class of potentials. Namely, we also assume that for any r ∈ R and some positive constants δ 0 and c 0 , c 1 , c 2 . We have written both the last two conditions in (2.25) for convenience even though the latter implies the former. We also remark that the classical potential (2.8) fulfils such assumptions. Here is our result.
Remark 2.7. We observe that even further regularity for σ could be derived on account of the regularity of R induced by (2.37), provided that σ 0 is smoother. It must be pointed out that a uniqueness result for the limit problem has been proved in [11] by a different argument. In the same paper, a slightly different regularity result is shown as well. Finally, we remark that the uniqueness property implies that the whole family (µ α , u α , σ α ) converges to (µ, u, σ) in the sense of (2.26)-(2.29) as α 0.
The rest of the section is devoted to list some facts and to fix some notations. We recall that Ω is bounded and smooth. So, throughout the paper, we owe to some wellknown embeddings of Sobolev type, namely V ⊂ L p (Ω) for p ∈ [1,6], together with the related Sobolev inequality In (2.39), C only depends on Ω, while C p in (2.40) also depends on p. In particular, the continuous embedding W ⊂ W 1,6 (Ω) ⊂ C 0 (Ω) holds. Some of the previous embeddings are in fact compact. This is the case for V ⊂ L 4 (Ω) and W ⊂ C 0 (Ω). We note that also the embeddings W ⊂ V , V ⊂ H and H ⊂ V * are compact. Moreover, we often account for the well-known Poincaré inequality where C depends only on Ω. Furthermore, we repeatedly make use of the notation and of well-known inequalities, namely, the Hölder inequality and the elementary Young inequality: Next, we introduce a tool that is generally used in the context of problems related to the Cahn-Hilliard equations. We define N : dom N ⊂ V * → V as follows: and, Note that problem (2.45) actually has a unique solution w ∈ V since Ω is also connected and that w solves the homogeneous Neumann problem for −∆w = v * in the special case v * ∈ H. It is easily checked that Inequality (2.48) (where one could check that C = 1 actually is suitable) essentially says that · * and the usual dual norm · V * are equivalent on dom N (the opposite inequality is straightforward, indeed). Finally, throughout the paper, we use a small-case italic c without any subscript for different constants, that may only depend on Ω, the constant γ, the shape of the nonlinearities p, β and λ, and the norms of the initial data related to assumption (2.11). We point out that c does not depend on α nor on the final time T nor on the parameter ε we introduce in a forthcoming section. For any parameter δ, a notation like c δ or c(δ) signals a constant that depends also on the parameter δ. This holds, in particular, if δ is either α, or T , or the pair (α, T ). The reader should keep in mind that the meaning of c and c δ might change from line to line and even in the same chain of inequalities, whereas those constants we need to refer to are always denoted by different symbols, e.g., by a capital letter like in (2.39) or by a letter with a proper subscript as in (2.25).

Uniqueness
In this section, we prove the uniqueness part of Theorem 2.2, that is, we pick two solutions (µ i , u i , σ i , ξ i ), i = 1, 2, and show that they are the same. As both α and T are fixed, we avoid to stress the dependence of the constants on such parameters. Moreover, as the solutions we are considering are fixed as well, we can allow the values of c to depend on them, in addition. So, we write (2.19) and some of the equations of (2.14)-(2.18) for both solutions and take the difference. If we set µ := µ 1 − µ 2 for brevity and analogously define u, σ and ξ, we have We note that (3.1) implies Ω (αµ + u + σ)(t) = Ω (αµ + u + σ)(0) = 0 for every t, so that We note that the term involving ξ is nonnegative since β is monotone. So, we rearrange and estimate the right-hand side we obtain by accounting for (2.48) and the Lipschitz continuity of λ as follows for every δ > 0. Next, we test (3.3) by σ and get since p is Lipschitz continuous and bounded. Now, we estimate the last two integrals, separately. By the regularity (2.12) of µ 2 and σ 2 and the Sobolev inequality (2.39), we have At this point, we combine the last two estimates with (3.5) and sum to (3.4). Then, we take δ small enough in order to absorb the corresponding terms in the left-hand side. By applying the Gronwall lemma, we obtain µ = 0, u = 0 and σ = 0. By comparison in (3.2) we also deduce ξ = 0, and the proof is complete.

A priori estimates
In this section, we introduce an approximating problem and prove a number of a priori estimates on its solution. Some of the bounds we find may depend on α and T , while other ones are independent of such parameters. The notation we use follows the general rule explained at the end of Section 2. The estimates we obtain will be used in the subsequent sections in order to prove our results.
For ε ∈ (0, 1), the approximation to problem (2.12)-(2.18) is obtained by simply replacing (2.13) by ξ = β ε (u) (4.1) where β ε and the related functions β ε and W ε are defined on the whole of R as follows It turns out that β ε is a well-defined C 1 function and that β ε , the Yosida regularization of β, is Lipschitz continuous. Moreover the following properties 0 ≤ β ε (r) ≤ β (r) and β ε (r) β (r) monotonically as ε 0 (4.3) hold true for every r ∈ R (see, e.g., [ Uniqueness is already proved as a special case of the uniqueness part of Theorem 2.2. As far as existence is concerned, we avoid a detailed proof and just say that a Faedo-Galerkin method (obtained by taking a base of V , e.g., the base of the eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions) and some a priori estimates very close to the ones we are going to perform in the present section lead to the existence of a solution. We also remark that the system of ordinary differential equations given by the Faedo-Galerkin scheme has a unique global solution since β ε is Lipschitz continuous and the function (µ, u, σ) → p(u)(σ − γµ) on R 3 is smooth (since p is so) and sublinear (since p is bounded).
From now on, (µ, u, σ) = (µ ε , u ε , σ ε ) stands for the solution to the approximating problem. Accordingly, we define R ε by (2.14). However, we explicitely write the subscript ε only at the end of each calculation, for brevity.

Existence on a finite time interval
In this section, we conclude the proof of Theorem 2.2 by showing the existence of a solution. As α and T are fixed now, we can account for all of the estimates of the previous section. Owing to standard weak compactness arguments as well as by the strong compactness result in [18,Sect. 8,Cor. 4], it turns out that the following convergence holds at least for a subsequence. The strong convergence in C 0 ([0, T ]; H) entails that the limiting functions satisfy the initial conditions (2.18). Moreover, the pointwise convergence a.e. and assumption (2.4) imply that R ε converges to R := p(u)(σ − γµ) a.e. in Q, so that ρ = R. Furthermore, as λ is Lipschitz continuous, λ (u ε ) converges to λ (u) strongly in L 2 (0, T ; H). Finally, a standard monotonicity argument (see, e.g., [1, Lemma 1.3, p. 42]) based on the weak convergence u ε → u, β ε (u ε ) → ξ in L 2 (Q) and on the property (easily following from (5.1)-(5.2)) yields ξ ∈ β(u) a.e. in Q. Therefore, the quadruplet (µ, u, σ, ξ) satisfies (2.12)-(2.13) and solves (2.14)-(2.18).
We conclude this section by recovering the uniform estimates for the solution (µ, u, σ, ξ) to the problem (2.12)- (2.18). First of all, we can speak of a unique solution on the time half line [0, +∞). For such a solution, the estimates we found for the approximating problem still hold, i.e., we have the last one for every T ∈ (0, +∞). This simply follows from the weak semicontinuity of the norms for all the inequalities but the one involving W(u). As far as the latter is concerned, we note that, for all t ≥ 0, u ε (t) → u(t) strongly in H. Then, using (2.6) and the mean value theorem, it is not difficult to check that and consequently λ(u ε (t)) converges to λ(u(t)) strongly in L 1 (Ω). Hence, in view of (2.7) and (4.2) it suffices to prove that Ω β (u(t)) ≤ lim inf ε 0 Ω β ε (u ε (t)). (5.6) To this end, we fix ε > 0 for a while. By accounting for the lower semicontinuity of β ε and the inequality β ε (s) ≤ β ε (s) which holds for every s ∈ R and ε ∈ (0, ε ) (see (4.2)), we obtain Now, we let ε vary and recall (4.3) in terms of ε . Thus, the Beppo Levi monotone convergence theorem implies that

Long time behavior
In this section, we prove Theorem 2.4. From (5.4) we see that the omega limit ω we are interested in is non-empty. It remains to characterize its elements as in the statement. So, we fix (µ ω , u ω , σ ω ) ∈ ω and a sequence {t n } according to definition (2.22). We set for convenience v n (t) := v(t + t n ) for t ≥ 0 with v = µ, u, σ, ξ, R (6.1) and study the behavior of such functions in a fixed finite time interval (0, T ). First of all, we notice that the quadruplet (µ n , u n , σ n , ξ n ) solves the problem obtained from problem (2.14)-(2.18) by replacing the initial condition (2.18) by the following one µ n (0) = µ(t n ), u n (0) = u(t n ) and σ n (0) = σ(t n ).
Therefore, at least for a subsequence, we also have µ n → µ ∞ , u n → u ∞ and σ n → σ ∞ weakly star in L ∞ (0, T ; V ) ∩ L 2 (0, T ; W ) (6.3) µ n → µ ∞ u n → u ∞ and σ n → σ ∞ strongly in L 2 (0, T ; H) (6.4) ξ n → ξ ∞ weakly in L 2 (0, T ; H) (6.5) the strong convergence (6.4) being a consequence of (6.3) and of the bounds for the time derivatives. In particular, thanks to the Lipschitz continuity of p, we derive that R n converges to p(u ∞ )(σ ∞ − γµ ∞ ) strongly in L 1 (Q), whence Furthermore, µ ∞ and σ ∞ are constant functions and u ∞ is time independent. We denote the constant values of µ ∞ and σ ∞ by µ s and σ s , respectively, and set u s := u ∞ (t) for t ∈ (0, T ). By taking the limit in (2.15) written for µ n and u n , we see that the pair (u ∞ , ξ ∞ ) solves the following problem a.e. in Q and ∂ ν u ∞ = 0 a.e. on Σ.
Remark 6.1. Even though we have to confine ourselves to study the omega limit of an initial datum satisfying (2.11), we could take a phase space Φ that is larger than (2.21) and is endowed with a weaker topology. This may lead to further properties of ω. For instance, if we choose Φ = (L 2 (Ω)) 3 with the strong topology, estimate (5.3) implies that the whole trajectory of the initial datum is relatively compact in Φ, so that general results (see, e.g., [19, Lemma 6.3.2, p. 239]) ensure that ω is invariant, compact and connected in the L 2 topology.

Asymptotics and limit problem
In this section, we perform the proof of Theorem 2.6. As T is fixed, we avoid stressing the dependence of the constants on T .
i) As in the statement, (µ α , u α , σ α , ξ α ) (where ξ α = β(u α ) since β is smooth) is the solution to problem (2.14)-(2.18) and we define R α accordingly. We recall that (5.3) implies u α L ∞ (0,T ;V ) ≤ c, whence also u α L ∞ (0,T ;L 6 (Ω)) ≤ c due to the Sobolev inequality (2.39). By the assumption (2.25), we infer that L ∞ (0,T ;L 6 (Ω)) + 1 ≤ c . (7.1) Now, we integrate (2.15) over Ω and use the homogeneous Neumann boundary condition for u α . Then, we square and integrate over (0, T ) with respect to time. We obtain the last inequality following from (5.3) and (7.1). Then, recalling the estimate for ∇µ α in (5.3) and owing to the Poincaré inequality (2.41), we conclude that By comparison in (2.15), we deduce ∆u α L 2 (0,T ;H) ≤ c, whence also u α L 2 (0,T ;W ) ≤ c (7.3) by elliptic regularity. Now, we test (2.14) by an arbitrary v ∈ L 2 (0, T ; V ) and get by the estimate (5.3) of R α and (7.2). This means that At this point, we can use weak and weak star compactness and conclude that at least for a subsequence. This proves the part i) of the statement but (2.29), which is more precise than (7.9) and is justified in the next step.
On the other hand αµ α → 0 in L 2 (0, T ; V ) by (7.6). Hence u α → u strongly in L 2 (0, T ; H) and ζ = ∂ t u. (7.10) By a standard argument (the same as in Section 5), we can identify the limit of R α as p(u)(σ − γµ) and the limits of the other nonlinear terms. Thus, we conclude that (µ, u, σ) solves (2.30)-(2.34) (in fact, one proves an equivalent integrated version of (2.30) rather than (2.30) itself). It remains to check the first condition in (2.35). By also accounting for (7.9), we see that αµ α + u α converges to u weakly in C 0 ([0, T ]; V * ). This implies that On the other hand, αµ 0 + u 0 → u 0 strongly in V . Therefore, u(0) = u 0 , and the proof of ii) is complete.
iii) A formal estimate that leads to (2.37)-(2.38) could be obtained by testing (2.30) by ∂ t u, differentiating (2.32) with respect to time, testing the obtained equality by µ and adding up. Here we perform the correct procedure, namely, the discrete version of the formal one, by introducing a time step h ∈ (0, 1). For simplicity, we allow the (variable) value of the constant c to depend on the norm u 0 W involved in (2.36) and on the solution we are considering (which is fixed). Of course, c does not depend on h. First of all, we introduce a notation. For v ∈ L 2 (−1, T ; H) and h ∈ (0, 1), we define the mean v h ∈ L 2 (0, T ; H) and the difference quotient δ h v ∈ L 2 (0, T ; H) by setting for t and we do the same if v ∈ (L 2 (−1, T ; H)) 3 in order to treat gradients. We notice that v h L 2 (0,T ;H) ≤ v L 2 (−1,T ;H) (7.12) as we show at once. We have indeed As we are going to apply (7.12) to µ and R, we need to extend such functions to the whole of Ω × (−1, T ). Our tricky construction is based on assumption (2.36) on u 0 and involves u as well. We first solve a backward variational problem with the help of the theory of linear abstract equations. We set and construct the Hilbert triplet (W 0 , H 0 , W * 0 ), where W * 0 is the dual space of W 0 , by embedding H 0 into W * 0 in the standard way. In the sequel, the symbol · , · denotes the duality pairing between W * 0 and W 0 . We introduce the continuous bilinear form a : W 0 × W 0 → R by setting a(z, v) := Ω (∆z) (∆v) for z, v ∈ W 0 and observe that a(v, v) + v 2 H ≥ α v 2 W for some α > 0 and every v ∈ W 0 , thanks to the elliptic regularity theory. We also notice that ∆u 0 ∈ H 0 since u 0 ∈ W . Therefore, as is well known (e.g., [17,Prop. 2.3 p. 112]), there exists a unique z satisfying (t), v) = 0 for every v ∈ W 0 and for a.a. t ∈ (−1, 0) (7.14) z(0) = −∆u 0 (7.15) and we also have As z ∈ C 0 ([−1, 0]; H 0 ), for every t ∈ [−1, 0] we have that z(t) ∈ dom N (see (2.44)). Hence, we can define a function w by setting w(t) := N(z(t)) for every t ∈ [−1, 0] (7.17) and it turns out that w ∈ C 0 ([−1, 0]; W ): the restriction of N to H 0 is an isomorphism from H 0 onto W 0 , indeed. Moreover, w is even smoother. Namely, from (7.16) we have that w L ∞ (−1,0;W ) ≤ c and ∂ t w L 2 (−1,0;H) ≤ c . (7.18) Here, the former is due to the above argument and we now prove the latter. Clearly, an estimate on the difference quotients is sufficient to conclude. We observe that the operator −∆ : W 0 → H 0 is a well-defined isomorphism. Thus, the same property is enjoyed by its adjoint operator (−∆) * : H 0 → W * 0 given by Hence, for every w * ∈ W * 0 there exists a unique y ∈ H 0 such that (−∆) * y = w * , i.e., Ω y(−∆v) = w * , v for every v ∈ W 0 (7. 19) and the estimate y H ≤ C w * W * 0 holds true with C depending only on Ω. Assume now h ∈ (0, 1) and t ∈ (−1+h, 0). From the definition (7.17) of w we immediately derive that y = δ h w(t) belongs to H 0 and satisfies i.e., it fulfils (7.19) with w * = δ h z(t). Therefore, we have δ h w(t) H ≤ c δ h z(t) W * 0 for every t ∈ (−1+h, 0), and (7.16) immediately implies This proves the second estimate in (7.18). Once (7.18) is completely established, we go on. We term (u 0 ) Ω the mean value of u 0 and notice that u 0 − (u 0 ) Ω coincides with w(0) given by (7.17) and (7.15). Therefore, we are suggested to define u in (−1, 0) by setting u(t) := w(t) + (u 0 ) Ω for t ∈ (−1, 0). (7.20) By doing that, we have both the estimates u L ∞ (−1,0;W ) ≤ c and ∂ t u L 2 (−1,0;H) ≤ c (7.21) and the fact that u(t) → u 0 (e.g., in H) as t 0. This implies that the extended function u ∈ L 2 (−1, T ; H) (which is continuous in [−1, 0] and [0, T ], separately) does not jump at t = 0. Thus, u ∈ C 0 ([−1, T ]; H) and its time derivative ∂ t u is a V * -valued function (rather than a distribution) on the whole of (−1, T ), so that (2.30) will hold in the whole of (−1, T ) whenever we properly extend µ and R. As the former is concerned, we set µ(t) := z(t) + W (u(t)) = −∆u(t) + W (u(t)) for t ∈ (−1, 0). (7.22) Now, in order to properly extend R, we check that ∆µ is a well defined function. Indeed, ∆z ∈ L 2 (−1, 0; H) by (7.16). On the other hand ∆W (u) = W (u)|∇u| 2 + W (u)∆u ∈ L ∞ (−1, 0; H) on account of (7.21), assumption (2.25) on W and the Sobolev embedding W ⊂ W 1,4 (Ω). Hence, we can set We obtain µ L 2 (−1,0;W ) ≤ c and R L 2 (−1,0;H) ≤ c .