$L_p$-theory for a Cahn-Hilliard-Gurtin system

In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.


Introduction
We start with the derivation of the classical Cahn-Hilliard equation. Consider the free energy functional of the form where Ω is a bounded, open and connected subset of R n with boundary Γ := ∂Ω ∈ C 3 . We assume that the order parameter ψ is a conserved quantity. The according conservation law reads where j is a vector field representing the phase flux of the order parameter. The next step is to combine the two quantities j and µ. Similar to Fourier's law in the derivation of the heat equation one typically assumes that j is given by a postulated relation. Finally we have to derive an equation for µ. The chemical potential µ is given by the variational derivative of F , i.e.
If F is of the form (1.1) this yields the classical Cahn-Hilliard equation.
In the early nineties Gurtin [8] proposed a generalized Cahn-Hilliard equation, which is based on the following objections: • Fundamental physical laws should account for the work associated with each operative kinematical process; • There is no clear separation of the balance law (1.2) and the constitutive equation (1.3); • Forces that are associated with microscopic configurations of atoms are not considered in the derivation of the classical Cahn-Hilliard equation. According to Gurtin there should exist so called 'microforces' whose work accompanies changes in the order parameter ψ. The microforce system is characterized by the microstress ξ ∈ R n and scalar quantities π and γ which represent internal and external microforces, respectively. The main assumption in [8] is that ξ, π and γ satisfy the (local) microforce balance (1.4) div ξ + π + γ = 0, which can be motivated from a static point of view, see [8] for more details. In a next step we want to derive constitutive equations, which relate the quantities j, the flux of the order parameter, ξ and π to the fields ψ and µ. The technique used in [8] for this derivation is based on the balance equation (1.4) and a (local) dissipation inequality, which is a direct consequence of the first and the second law of thermodynamics, that is, the energy balance d dt Ω e dx = − ∂Ω q · ν dσ + Ω r dx + W(Ω) + M(Ω), cf. [8,Appendix A]. The second law of thermodynamics is also known as the Clausius-Duhem inequality. Here e is the internal energy, S is the entropy, θ is the absolute temperature, q is the heat flux, r is the heat supply, W(Ω) is the rate of working on Ω of all forces exterior to Ω and M(Ω) is the rate at which energy is added to Ω by mass transport. Let F be the free energy density, depending on the vector z = (ψ, ∇ψ, µ, ∇µ, ∂ t ψ). Then the second law of thermodynamics (in its mechanical version as considered by Gurtin [8]) reads with m being the external mass supply. Making use of Green's formula, we obtain in presence of external mass supply m, (1.2) will be modified to (1.5) ∂ t ψ + div j = m.
In Section 2, where we consider Ω = R n , we allow for general, positive definite matrices B. It is also possible to consider those matrices in all other sections but for the sake of convenience we restrict ourself to the case B = bI. Actually this allows to draw back the problem in the half space R n + to the whole space R n by means of reflection methods. Results on existence and uniqueness can be found e.g. in the papers of Bonfoh & Miranville [3], Miranville [10], [11], Miranville & Piétrus [16], Miranville, Piétrus & Rakotoson [12] and Miranville & Zelik [14]. In any of these papers the authors use a variational approach and energy estimates to obtain global wellposedness in an L 2 -setting, with periodic boundary conditions for a cuboid in R 3 . The qualitative behavior of solutions of the Cahn-Hilliard-Gurtin equation has been investigated in [3], [12] and [13]. In [3] and [12] the authors proved the existence of finite dimensional attractors, whereas Miranville & Rougirel [13] showed that each solution converges to a steady state, again with the help of the Lojasiewicz-Simon inequality. One assumption of Miranville & Rougirel [13] is that the norms |a|, |c| and |B − I| are bounded by a possibly small constant. In the present paper we will give an alternative proof for the relative compactness of the orbit {ψ(t)} t≥0 in H 1 2 (Ω) with the help of semigroup theory and a priori estimates (see Proposition 7.1). The present paper is structured as follows. In Section 2 we deal with a corresponding linearized system to (1.9) in the full space R n with constant coefficients. Section 3 is devoted to the analysis of the linearized system with constant coefficients in the half space R n + . Making use of the optimal regularity results of Sections 2 and 3 we apply the method of localization and some perturbation results in Section 4 to derive optimal L p -regularity for the linearized Cahn-Hilliard-Gurtin equations (i.e. (1.8) with Φ ′ = 0) in an arbitrary bounded domain Ω ⊂ R n with boundary ∂Ω ∈ C 3 . In Section 5 we prove the existence and uniqueness of a local-in-time solution of (1.9). For this purpose it is crucial to have the optimal L p -regularity result from Section 4 at our disposal. To the knowledge of the author there are no results on the local well-posedness of (1.9) but only for the case where a, c and b depend solely on the order parameter ψ, cf. Miranville [15]. In Section 6 we investigate the global wellposedness of the semilinear system (1.8). The basic tools are a priori estimates and the Gagliardo-Nirenberg inequality. Finally, in Section 7, we show that each solution ψ(t) of (1.8) converges to a steady state in H 1 2 (Ω) as t → ∞. To this end we will use relative compactness results and the Lojasiewicz-Simon inequality.

The Linear Cahn-Hilliard-Gurtin Problem in R n
In this section we will solve the full space problem where β > 0, a, c ∈ R n and B ∈ R n×n . Note that the matrix (1.7) is positive semidefinite if and only if βz 2 0 + (a + c|z 1 )z 0 + (Bz 1 |z 1 ) ≥ 0 holds for all (z 0 , z 1 ) ∈ R × R n and all x ∈ Ω. Here (·|·) denotes the usual scalar product in C n and the vector fields a, c as well as the matrix valued function B are assumed to be smooth. In the sequel we will use a slightly stronger assumption.
(H) There is a constant ε > 0, such that is valid for all (z 0 , z 1 ) ∈ R × R n and all x ∈ Ω.
The following result is useful for the analysis of (2.1) (see also [13,Lemma 5.1]).
Proposition 2.1. Let (H) hold. Then Proof. Hypothesis (H) reads Observe that the left side of this inequality can be rewritten as For a fixed z 1 ∈ R n we choose z 0 ∈ R in such a way that the squared bracket is equal to 0. Thus we obtain the estimate valid for all z 1 ∈ R n . By the definition of d it holds that hence we obtain the identity Since the matrix (a−c)⊗(a−c) is positive semi-definite we finally obtain the assertion.
Here is the main result on optimal L p -regularity of (2.1).
Theorem 2.2. Let 1 < p < ∞ and assume that (H) holds true. Then (2.1) admits a unique solution µ ∈ L p (J; H 2 p (R n )) =: Z 2 , if and only if the data is subject to the following conditions.
Observe that the converse is also true, i.e. there is a constant C > 0 such that In particular it holds that m(λ, ξ) = 0 if and only if |λ| + |ξ| = 0.
The existence of v 0 and v 1 may be seen by the Dore-Venni-Theorem. It follows that where the linear operator S is defined by its Fourier-Laplace symbol Note that the assertion of Theorem 2.2 follows if we can show that S is a bounded operator from L p (J; L p (R n )) to L p (J; L p (R n )). This will be a consequence of the classical Mikhlin multiplier theorem and the Kalton-Weis Theorem [9,Theorem 4.5].
It is not difficult to show that the symbolŜ(λ, ξ) satisfies the Mikhlin condition where α ∈ N n 0 is a multiindex and [s] denotes the largest integer not exceeding s ∈ R. The classical Mikhlin multiplier theorem then implies thatŜ is a Fourier multiplier in L p (R n ; C) w.r.t. the variable ξ and this yields a holomorphic uniformly bounded family {S(λ)} λ∈Σ φ ⊂ B(L p (R n ; C)), φ > π/2. By [7, Theorem 3.2] this family is also R-bounded in L p (J; L p (R n ; C)) (for the notion of R-boundedness we refer the reader to [5]). Finally, since the operator ∂ t admits a bounded H ∞ -calculus with angle π/2 we obtain from [9, Theorem 4.5] the desired property of the operator S. For the functions u = (I − ∆) 1/2 w and µ = (I − ∆) 1/2 η, this yields , as well as µ ∈ L p (J; H 2 p (R n )), and the proof is complete.
For later purpose we need a perturbation result. To be precise we consider coefficients a, c and B with a small deviation from constant ones, i.e.
Furthermore we assume that div a 1 (x) = div c 1 (x) = 0 for a.e. x ∈ R n and that the quadruple (β, a 0 , c 0 , B 0 ) satisfies (H). Observe that if ω > 0 is sufficiently small, then (β, a(x), c(x), B(x)) satisfy (H) as well for all x ∈ Ω, with a possibly smaller constant ε > 0. We have the following result. Proof. By a shift of the function u we may assume that u 0 = g = 0. For the time being we consider an interval J δ = [0, δ], with a suitable small δ > 0, to be chosen later. The corresponding function spaces are denoted by X j δ and Z j δ . Moreover 0 Z 1 δ := {u ∈ Z 1 δ : u| t=0 = 0}. Assume that we already know a solution (u, µ) ∈ 0 Z 1 δ × Z 2 δ of (2.1). Thanks to Theorem 2.2 we have a solution operator S ∈ B(X 1 δ × X 2 δ × X p ; 0 Z 1 δ × Z 2 δ ) for the constant coefficient case (β, a 0 , c 0 , B 0 ). With the help of S we write the solution in the following way.
From the boundedness of S and since div a 1 (x) = 0, x ∈ R n , we obtain the estimate (2.5) for some constant C > 0. The problem is that the term |∇µ| does not become small in L p (J δ ; L p (R n )), since the function µ has no regularity w.r.t. the variable t. However, we have the following result.
Proposition 2.4. Let (u, µ) ∈ Z 1 δ × Z 2 δ be a solution of (2.1) with g = u 0 = 0. Assume furthermore that the (variable) coefficients satisfy the above assumptions. Then there exists a constant C > 0, independent of J δ , such that the estimate Proof. The proof follows the lines of the proof of Proposition 3.3.
Owing to (2.5) and Proposition 2.4 we obtain the estimate The mixed derivative theorem and Sobolev embedding yield , hence by Hölder's inequality we obtain |u| Lp(J δ ;H 2 p (R n )) ≤ δ 1/2p C|u| Z 1 δ and the constant C > 0 does not depend on δ > 0, since u| t=0 = 0. Choosing first ω > 0, then δ > 0 small enough and shifting back the function u, we obtain from (2.7) the estimate is injective and has closed range, hence L is a semi-Fredholm operator. Replacing the coefficients (β, a, c, B) by we may conclude from the considerations above that for each τ ∈ [0, 1] the corresponding operator L τ is semi-Fredholm as well. The continuity of the Fredholm index yields that the index of L 1 = L is 0, since L 0 is an isomorphism, by Theorem 2.2. A successive application of the above procedure yields the claim for the time interval J = [0, T ]. The proof is complete.
3. The Linear Cahn-Hilliard-Gurtin Problem in R n + In order to treat the case of a half space, we consider first constant coefficients which are subject to the following assumptions: B = bI and (a|e n ) = (c|e n ) = 0, where e n := [0, . . . , 0, −1] T is the outer unit normal at ∂R n + . Furthermore we assume that (β, a, c, B) satisfy (H), whence it holds that b ≥ ε > 0. Moreover the boundary conditions on a and c yield that the last components of a and c are identically zero. We are interested to solve the following system in R n + .
Hence it suffices to consider the boundary condition ∂ y µ = h 1 with some scaled function h 1 . Concerning optimal L p -regularity of (3.1) we have the following result.
Theorem 3.1. Let 1 < p < ∞, p = 3/2 and assume that (H) holds true. Then (3.1) admits a unique solution µ ∈ L p (J; H 2 p (R n + )) =: Z 2 , if and only if the data is subject to the following conditions. ( Proof. The necessity part follows from the equations and trace theory, cf. [6]. Concerning sufficiency, we first reduce (3.1) to the case h 1 = h 2 = u 0 = 0. For this purpose we solve the elliptic problem and let L denote the natural extension ofL to L p,loc (R + ; L p (R n−1 )), that is D(L) = L p,loc (R + ; H 1 p (R n−1 )) and Lu =Lu for each u ∈ D(L). Then the unique solution η of (3.2) is given by we have e −Ly h 1 ∈ D(L) and therefore η ∈ L p (J; H 2 p (R n + )), with ∂ y η| y=0 = h 1 . In order to remove h 2 and u 0 , we solve the initial boundary value problem and solve the heat equation ). Then, the unique solution v 3 of (3.5) is given by ). On the other hand, if we consider the function v 4 := ∂ y v 2 as the solution of . From the regularity of v 3 and v 4 we may conclude that with some modified data f 1 ∈ X 1 and g 1 ∈ X 2 . In a next step we extend the functions f 1 and g 1 w.r.t. the spatial variable to R n by even reflection, i.e. we set Thanks to Theorem 2.2 we can solve the full space problem ). This yields a unique solution , by Theorem 2.2. At this point we emphasize that the equations (3.7) 1,2 are invariant w.r.t. even reflection on the hyper surface R n−1 × {0} in the normal variable y, due to the structure of the coefficients. This in turn implies that the solution (u 2 , µ 2 ) is symmetric, w.r.t the variable y and this yields necessarily, ∂ y u 2 | y=0 = ∂ y µ 2 | y=0 = 0. Denoting by P the restriction of the solution (u 2 , µ 2 ) to the half space R n + , it follows that (u 1 , µ 1 ) = P (u 2 , µ 2 ) is the unique solution of (3.7) and therefore u = v + u 1 and µ = η + µ 1 is the unique solution of (3.1). The proof is complete.
For later purposes we will need the following perturbation result.
If the constant coefficients (β, a 0 , c 0 , B 0 ) satisfy Hypothesis (H) we have the following result.
Proof. First of all, we reduce (3.9) to the case u 0 = 0 as follows. Extend the initial and solve the heat equation Observe thatf ,g andh 2 depend only on f, g, h 2 and the fixed function v ∈ Z 1 from above. In the sequel we will not rename the functions u, f, g and h 2 . By the structure of the coefficients and by trace theory we obtain the estimate , with a constant C > 0 which does not depend on δ > 0 since u| t=0 = 0. The derivation of this estimate follows the lines of the proof of Corollary 2.3. The term |u| Lp(J δ ;H 2 p (R n + )) is of lower order and may be estimated by |u| Lp(J δ ;H 2 δ , hence this term may be compensated by the left side of the latter estimate if δ > 0 is small enough. If in addition ω > 0 is sufficiently small, the same is true for To estimate the term |∇µ| in L p (J δ ; L p (R n + )), we use the following proposition whose proof is given in the Appendix.
δ be a solution of (3.9) with u 0 = 0. Then there exists a constant C > 0, independent of J δ , such that the estimate Now the claim follows by applying a similar homotopy argument as in the proof of Corollary 2.3.
and Ω ⊂ R n be a bounded domain with ∂Ω ∈ C 3 . Then for each β > 0 the initial-boundary value problem admits a unique solution , if and only if the data are subject to the following conditions. ( Proof. The 'only if' part follows from the equations and well known result in trace theory. Indeed, given a solution by trace-and interpolation theory.
. Taking the trace of ∇u on ∂Ω yields the required regularity for g. Finally, since it follows that ∂ ν u(0) = g| t=0 in case p > 3/2. To prove sufficiency of the conditions (i)-(iv), note that by the results of Sections 2 & 3 the unique solution of the corresponding full space and half space problem to (4.2) possess the desired regularity.
Then the claim for a bounded domain Ω ⊂ R n with ∂Ω ∈ C 3 follows from localization, change of coordinates and perturbation theory, cf. [5].
The second lemma provides maximal regularity of (4.1) in case a = c = 0 and b = 1, the so-called viscous Cahn-Hilliard equation in its linear form.
and Ω ⊂ R n be a bounded domain with ∂Ω ∈ C 3 . Then for each β > 0 the system admits a unique solution , if and only if the data are subject to the following conditions. ( Hence, w.l.o.g. we may assume g = h 2 = u 0 = 0 in (4.2), with f being replaced by some modified functionf ∈ L p (J; L p (Ω)), which depends at most on the fixed functions f and v. Now we want to reduce (4.3) to a single equation for u. Suppose that we already know a solution of (4.3). Inserting (4.3) 1 into (4.3) 2 yields the elliptic problem for the function µ. It is well-known that for each β > 0 the latter problem admits a unique solution µ ∈ L p (J; H 2 p (Ω)), provided (βf − ∆u) ∈ L p (J; L p (Ω)) and h 1 ∈ L p (J; W 1−1/p p (∂Ω)). Denoting by S the corresponding solution operator, we may write Inserting this expression into (4.3) 2 we obtain the problem ) it follows thatS is bounded and linear from L p (J; H 2 p (Ω)) to L p (J; H 2 p (Ω)). Thanks to Lemma 4.1 there exists a solution operator T of (4.2) which is a linear and bounded mapping from . With the help of T we may write We estimate Proof. By Lemma 4.2 we may first reduce (4.1) to the case h 1 = h 2 = u 0 = 0 and some modified functions f, g in the right regularity classes. We cover Ω by finitely many open sets U k , k = 1, ..., N , which are subject to the following conditions.
We choose next a partition of unity {ϕ k } N k=1 such that N k=1 ϕ k (x) = 1 on Ω, 0 ≤ ϕ k (x) ≤ 1 and supp ϕ k ⊂ U j . Note that (u, µ) is a solution of (4.1) if and only if Here we have set u k = uϕ k , µ k = µϕ k , f k = f ϕ k , g k = f ϕ k . The terms F k (u, µ) and G k (u, µ) are defined by

and
G k (u, µ) = −(c · ∇ϕ k )µ + 2∇u∇ϕ k + u∆ϕ k . In case k = 1, ..., N 1 we have no boundary conditions, i.e. we only have to consider the first two equations in (4.5). In order to treat these local problems with the help of Corollary 2.3 we extend the coefficients from B r k (x k ) to R n in such a way that divã(x) = divc(x) = 0, x ∈ R n , holds for the extended coefficientsã andc. Note that w.l.o.g. we may assume x k = 0. This follows by a translation in R n . We use the following extensionã of a (orc of c).
where r = |x|, ξ = x/|x| and ξ j , a j denote the components of ξ and a, respectively. The task is to compute the scalar valued function R(r, ξ). Since div a(x) = 0, x ∈ Ω, the divergence of a The divergence of the last term R(r, ξ)ξ is given by Finally, this yields that divã k (x) = 0 if and only if the function R = R(r, ξ) solves the ordinary differential equation In order to achieveã k ,c k ∈ W 1 ∞ (R n ; R n ), we requireã k (r k ξ) = a(r k ξ). This yields the initial condition R(r k , ξ) = 2(a(r k ξ)|ξ), hence the function R = R(r, ξ) is explicitly given by R(r, ξ) = r n−1 k r n−1 R(r k , ξ) + 2(n − 1) r n−1 r r k s n−2 a r 2 k s ξ ξ ds, r ≥ r k .
Since 2 r n−1 k r n−1 (a(r k ξ)|ξ) = 2(a(r k ξ)|ξ) − 2 (n − 1) r n−1 r r k s n−2 (a(r k ξ)|ξ) ds, we may writẽ in case |x| > r k . Owing to this identity and the assumption a, c ∈ C 1 (Ω), it is evident that there holds where ω > 0 can be made as small as we wish, by decreasing the radius r k of the charts U k , k ∈ {1, ..., N 1 }. For the coefficient function b we use the reflection method from [5], i.e. we set It may be readily checked thatb k ∈ W 1 ∞ (R n ) and that with the same ω > 0 as above. Hence for each chart U k , k ∈ {1, ..., N 1 } we have coefficients which fit into the setting of Corollary 2.3. Therefore we obtain corresponding solution operators S F k of (4.5) such that for each k ∈ {1, . . . , N 1 }. For the remaining charts U k , k ∈ {N 1 +1, . . . , N } we obtain problems in perturbed half spaces with inhomogeneous Neumann boundary conditions. For the further analysis we have to understand how to treat (4.1) in such a setting. To this end we fix a point x 0 ∈ ∂Ω and a chart U (x 0 ) ∩ ∂Ω = ∅. After a composition of a translation and a rotation in R n , we may assume that x 0 = 0 and ν(x 0 ) = [0, . . . , 0, −1] = e n . Consider a graph ρ ∈ C 3 (R n−1 ), having compact support, such that Note that by decreasing the size of the charts we may assume that |∇ x ′ ρ| ∞ is as small as we like, since ∇ x ′ ρ(0) = 0. For the time being, we only know that div a(x) = div c(x) = 0 for all x ∈ U (x 0 ) ∩ Ω. So we have to extend the coefficients a and c in a suitable way. To this end we first transform the crooked boundary U (x 0 ) ∩ ∂Ω to a straight line in R n−1 × {0}. This will be done with the help of a suitable transformation. Let u(x ′ , x n ) = v(g(x)) = v(x ′ , x n − ρ(x ′ )) and µ(x) = η(g(x)) = η(x ′ , x n − ρ(x ′ )) and B r0 (x 0 ) = g(U (x 0 )). Then the differential operators a · ∇u and c · ∇µ transform as follows.
and some functions (f, g, h 1 , h 2 ) ∈ X 1 ×X 2 ×Y 1 ×Y 2 such that h 2 | t=0 = 0. From the extension method above it follows that for all x ∈ R n + where we can choose ω > 0 arbitrarily small, by decreasing the radius r 0 > 0 of the ball B r0 (x 0 ) = g(U (x 0 )). Furthermore it holds that |D(x) − I| ≤ ω, x ∈ R n + , since we may choose |∇ρ| ∞ as small as we wish. An application of Corollary 3.2 yields a unique solution operator S H of (4.9), hence ΘS H is the corresponding solution operator for the chart U (x 0 ). At this point we want to remark that the function 1 + |∇ x ′ ρ| 2 is a multiplier for the spaces W 1−1/p p (R n−1 ) and W 2−1/p p (R n−1 ), since ρ ∈ C 3 (R n−1 ) has compact support. This above computation yields solution operators Θ k S H k for the charts U k , k ∈ {N 1 + 1, . . . , N }, hence we may write for each k ∈ {N 1 + 1, . . . , N }. Summing (4.8) and (4.10) over all charts U k , k ∈ {1, . . . , N }, we obtain is a partition of unity. By the boundedness of the solution operators we obtain the estimate (4.12)  Choosing δ > 0 sufficiently small, we obtain from (4.12) and Proposition 4.4 the estimate , for a solution of (4.1). This shows that the bounded operator L : is injective and has closed range, i.e. it is semi Fredholm. Here Y δ is defined by To show surjectivity, we apply again the Fredholm argument to the set of data (β, a τ , c τ , B τ ) = (1 − τ )(β, 0, 0, I n ) + τ (β, a, c, B), τ ∈ [0, 1].
The corresponding operators L τ are semi Fredholm by the above procedure and by Lemma 4.2 the operator L 0 is bijective. The continuity of the Fredholm index thus yields that the index of L 1 = L is 0 and therefore the operator L is bijective as well.
A successive application of the above arguments yields existence of a unique solution (u, µ) of (4.1) on an arbitrary bounded interval [0, T ]. This completes the proof of Theorem 4.3.

Local Well-Posedness
Let p > n + 2, f ∈ X 1 , g ∈ X 2 , h j ∈ Y j , j = 1, 2 and ψ 0 ∈ X p be given such that the compatibility condition ∂ ν ψ 0 = h 2 | t=0 is satisfied. In this section we consider the quasilinear system where Φ ∈ C 3− (R). Assume that we have given vector fields a, c ∈ C 1 (Ω; C 2− (R × R n ; R n )) and a scalar valued function b ∈ C 1 (Ω; C 2− (R × R n ; R)) such that Suppose furthermore that (β,ã,c,b) are subject to Hypothesis (H) for each x ∈ Ω. Observe that for p > n+2 we have ψ 0 ∈ X p = B

3−2/p pp
(Ω) ֒→ C 2 (Ω), henceã,c ∈ [C 1 (Ω)] n andb ∈ C 1 (Ω) and therefore the coefficients, frozen at ψ 0 , satisfy the assumptions in Theorem 4.3. Thanks to Theorem 4.3 we may define a pair of functions (u * , v * ) ∈ Z 1 × Z 2 as the unique solution of the linearized system : h 2 | t=0 = 0} and denote by | · | 1 and | · | 0 the canonical norms in E 1 and E 0 , respectively. We define a linear operator L : and a nonlinear function Considering L as an operator from 0 E 1 to 0 E 0 , we obtain from Theorem 4.3 that L is a bounded isomorphism and by the open mapping theorem L is invertible with bounded inverse L −1 . It is easily seen that (ψ, µ) := (u + u * , v + v * ) is a solution of (5.1) if and only if Consider a ball B r ⊂ 0 E 1 where r ∈ (0, 1] will be fixed later. Define a nonlinear operator by T (u, v) := L −1 G((u, v), (u * , v * )). To apply the contraction mapping principle we have to show that T B r ⊂ B r and that there exists a constant κ < 1 such that the contractive inequality holds for all (u, v), (ū,v) ∈ B r . The following proposition is crucial to prove the desired properties of the operator T .
With the help of Proposition 5.1 we are able to prove the desired properties of the operator T defined above. We first care about the contraction mapping property.
(i) f ∈ L p (J; L p (Ω)) = X 1 , 3. An inspection of the proof of Theorem 5.2 shows that the assumption p > n + 2 can be relaxed to p > (n + 2)/3 in the semilinear case, i.e. if (a, b, c) are independent of ψ and ∇ψ. Indeed, it remains to estimate the nonlinearity Φ ′ (ψ) in L p (0, T ; H 1 p (Ω)). However, in the sequel we will always assume the stronger condition p > n + 2.
Integrating (6.3) with respect to t and choosing δ > 0 small enough, we obtain together with (6.6) and (6.7) the estimate In order to treat the last double integral, we have to assume more regularity for the function h 2 . To be precise, we assume that h 2 ∈ H 1 p (J; L p (Γ)) ∩ L p (J; W 2−1/p p (Γ)) ֒→ C(J; L p (Γ)).
The following proposition provides some properties of the ω-limit set Proposition 7.2. Suppose that (ψ, µ) is a global solution of (7.1) and let Φ satisfy Hypotheses (6.4) and (7.2). Then the following statements hold.
Taking the limit t n k → ∞ we obtain where a : V × V → R is defined by a(u, v) = (∇u, ∇v) 2 and (·, ·) 2 denotes the scalar product in L 2 (Ω). Since Φ ′ (ψ ∞ ) ∈ L q (Ω) with q = 6/(γ + 2) it follows that ψ ∞ ∈ D(A q ) = {u ∈ H 2 q (Ω) : ∂ ν u = 0}, where A q is the part of the operator A in L q (Ω) which is induced by the form a(u, v). Observe that q > 6/5 by assumption, whence we may apply a bootstrap argument to conclude ψ ∞ ∈ H 2 2 (Ω) and ∂ ν ψ ∞ = 0 on Γ (recall that q > 1 may be arbitrarily large in case n ∈ {1, 2}). Going back to (7.5) we obtain for (t n k ) ր ∞ the identity for all functions ϕ ∈ V . This yields (iii) after integration by parts. To prove (iv) observe that by [17,Proposition 5.2] the functional E is twice continuously Fréchet differentiable and its first derivative is given by Integration by parts finally yields assertion (iv).
At this point we could simply refer to the paper of Miranville & Rougirel [13] to prove the main Theorem 7.4 below. However, for the sake of completeness we provide a proof of this result. The next proposition is the key for the proof of the convergence of the orbit {ψ(t)} t≥0 towards a stationary state as t → ∞. Proposition 7.3 (Lojasiewicz-Simon inequality). Let ϕ ∈ ω(ψ) and assume in addition to (6.4) and (7.2) that Φ is real analytic. Then there exist constants s ∈ (0, 1 2 ], C, δ > 0 such that Proof. This is Proposition 6.6 in [4].  Proof. Since each element ϕ ∈ ω(ψ) is a critical point of E, Proposition 7.3 implies that the Lojasiewicz-Simon inequality is valid in some neighborhood of ϕ ∈ ω(ψ). By Proposition 7.2 (ii) the ω-limit set is compact, hence there exists N ∈ N such that where B δj (ϕ j ) ⊂ V are open balls with center ϕ i ∈ ω(ψ) and radius δ i . Additionally in each ball the Lojasiewicz-Simon inequality is valid. It follows from Proposition 7.2 (i) and (ii) that the energy functional E is constant on ω(ψ), i.e. E(ϕ) = E ∞ , for all ϕ ∈ ω(ψ). Thus there exists an open set U ⊃ ω(ψ) and uniform constants s ∈ (0, A well-known result in the theory of dynamical systems sates that the ω-limit set is an attractor for the orbit {ψ(t)} t∈R+ . To be precise this means lim t→∞ dist(ψ(t), ω(ψ)) = 0 in V.
This implies that there exists some time t * ≥ 0 such that ψ(t) ∈ U for all t ≥ t * and thus the Lojasiewicz-Simon inequality holds for the solution ψ(t), i.e.
Define a function H : Then with (7.3) and (7.6) it holds that The first Fréchet derivative of E in V reads for all (u, h) ∈ V × V . Setting u = ψ(t) and making use of (7.1) 2 we obtain with the help of Hölder's inequality, Poincaré's inequality and integration by parts since div c(x) = 0, x ∈ Ω and (c(x)|ν(x)) = 0, x ∈ ∂Ω. Taking the supremum in (7.8) over all functions h ∈ V with norm less than 1 it follows that We insert this estimate into (7.7) to obtain Integrating this inequality from t * to ∞ it follows that |∂ t ψ(·)| 2 , |∇µ(·)| 2 ∈ L 1 (R + ), since H(t) > 0. This implies that the limit lim t→∞ ψ(t) =: ψ ∞ exists firstly in L 2 (Ω) but by relative compactness also in V . Finally, by Proposition 7.2 (iii) the limit ψ ∞ is a solution of the stationary problem (7.4). The proof is complete.
Actually this splitting shows that problem (8.1) is indeed elliptic by Assumption (H) and Proposition 2.1, provided ω > 0 is sufficiently small. Will will now proceed in several steps.
Step 1. In this first step we want to reduce (8.1) to the case of homogeneous boundary conditions B(x, D)µ = 0. Consider the elliptic problem with constant coefficients (8.2) λµ − div(B 0 ∇µ) = f, x ∈ R n + , (B 0 ∇µ|ν) = g, x ∈ ∂R n + , whereB 0 := βB 0 − 1 2 (a 0 ⊗ c 0 + c 0 ⊗ a 0 ) and λ ∈ R is a parameter. Note that (8.2) is an elliptic problem with a conormal boundary condition and constant coefficients. Thanks to Proposition 2.1 the matrixB 0 is positive definite. By well known results it follows that for each f ∈ L p (R n + ) and g ∈ W 1−1/p p (∂R n + ) problem (8.2) has a unique solution µ ∈ H 2 p (R n + ), provided λ > 0. We remind that the variable coefficients have a small deviation from the constant ones a 0 , c 0 , B 0 , i.e.
Set F = a div(D∇u) ∈ H 1 p (R n + ) and f = div F ∈ L p (R n + ). Then µ 1 ∈ H 2 p (R n + ) is a solution of the abstract equation µ 1 + A 0 µ 1 = f . We claim that this f can be identified with a linear functional in E −1/2 (for which we will write f again where we also made use of (F |ν) = (a|ν) div(D∇u) = 0. Since A −1/2 is the E −1/2realization of A 0 (hence an extension of A 0 ) with 1 ∈ ρ(−A −1/2 ) = ρ(−A 0 ) and since f = div(a∆u) ∈ E −1/2 , we obtain a constant C > 0 such that the estimate , for the solution µ 1 ∈ H 2 p (R n + ) of (8.5) is valid. From the estimates for µ 1 and µ 2 and the embedding H 2 p (R n + ) ֒→ H 1 p (R n + ), we obtain a constant C > 0 such that . In the case that the functions depend on the parameter t it follows that the estimate (8.7) |µ(t)| H 1 , holds for a.e. t ∈ J = [0, T ] where the constant C > 0 is uniform in t, since the coefficients of the differential operators considered above are independent of t as well. Taking the p-th power and integrating (8.7) with respect to t, we obtain (8.8) |µ| Lp(J;H 1 p (R n + )) ≤ C |f | Lp(J;(R n + )) + |g| Lp(J;H 1 p (R n + )) + |h 1 | Lp(J;W 1−1/p p (R n−1 )) + |u| Lp(J;H 2 p (R n + )) . Finally the estimate for ∂ t u in L p (J; L p (R n + )) follows from (3.1) 2 and (8.8). The proof is complete.