Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation

We consider a nonlocal reaction-diffusion equation with mass conservation, which 
was originally proposed by Rubinstein and Sternberg as a model for phase 
separation in a binary mixture. 
We study the large time behavior of the solution and show that it converges to a stationary solution as $t$ tends to infinity. We also evaluate the rate of convergence. In some special case, we show that the limit solution is constant.

1. Introduction. We consider the nonlocal initial value problem (P ) where Ω ⊂ IR N (N ≥ 1) is a connected bounded open set with smooth boundary ∂Ω; ∂ ν is the outer normal derivative to ∂Ω and This model is mass conserved, namely Problem (P ) was introduced by Rubinstein and Sternberg [29] as a model for phase separation in a binary mixture. Although this problem is a nonlocal problem, we can prove the existence of an invariant set. The main result of this paper concerns the large time behavior. We show that the solution converges to a stationary solution as t tends to infinity and evaluate the rate of convergence. In some special case, we show that the limit solution is constant.
In the general case, the main tool to study the large time behavior is a Lojasiewicz inequality that was first proposed by Lojasiewicz himself [24], [26]. He showed that all bounded solutions of gradient systems in IR N , which are systems of ordinary differential equations, converge to a stationary solution. This idea was subsequently developed in infinite-dimensional spaces for proving the convergence to steady state for bounded solutions of several local equations such as reaction-diffusion equations, wave equations and degenerate parabolic equations [11], [12], [13], [14], [20], [21] [30], [31]; let us also mention some results on nonlocal phase-field models [10], [23], and the book on gradient inequalities by Huang [18].
In the case where f is supposed to be nonincreasing on an interval containing the range of the initial function, instead of applying Lojasiewicz inequality, we show that the ω-limit set only contains a unique element. This follows from the monotony of f and the mass conservation property. This result is related to [4,Theorem 3.9,page 88] where the author studies the asymptotic behavior of solutions for parabolic equations with a monotone operator. In our case, although the operator is not monotone because of the nonlocal term, we overcome this difficulty by using the mass conservation property.
In [29], the authors consider Problem (P ) with f of bistable type; a typical example is f (s) = s − s 3 . In this paper, we assume that the function f is of the form f (s) = n i=0 a i s i , where n ≥ 1 is an odd number, a n < 0. (1.1) Constants s 1 , s 2 : We suppose that s 1 < s 2 are two constants such that Note that we can choose s 1 , s 2 such that s 1 is negative with large absolute value and s 2 is arbitrarily large.
Assumption on initial data: We will make the following hypotheses on the initial data: This paper is organized as follows: in Section 2, a result on existence, uniqueness and boundedness of solutions is presented. Section 3 is devoted to prove a version of the Lojasiewicz inequality. In Section 4, we apply the Lojasiewicz inequality to prove that as t → +∞, u(t) converges to a stationary solution, which we precisely compute in the case of one space dimension. The convergence rate is established in Section 5.
Proof. We take the duality product of the equation for u by 1 to obtain which implies (2.1).
is a solution of Problem (P ) and that Then Proof. For the purpose of contradiction, we suppose that there exists a first time t 0 > 0 such that u(x 0 , t 0 ) = s 1 or u(x 0 , t 0 ) = s 2 for some x 0 ∈ Ω. Without loss of generality, assume that u(x 0 , t 0 ) = s 2 . By the continuity of u and the definition of t 0 , we have s 1 ≤ u(x, t 0 ) ≤ s 2 for all x ∈ Ω, and u(x, t) < s 2 for all x ∈ Ω and 0 ≤ t < t 0 .

3)
and In order to prove Theorem 2.2, we need some technical lemmas.
Lemma 2.3. Let u 0 ∈ L 2 (Ω), g ∈ L p (Q T ) for some p ∈ (1, ∞) and let u be the solution of the time evolution problem   Then for each 0 < ε < T , there exists a positive constant C 0 (ε, Ω, T ) such that . Remark 1. If T = 1, then the constant C 0 depends only on ε and Ω.
Lemma 2.4. One has the following embedding Proof of Theorem 2.2. One can prove in a standard way (see also Proposition 1) the existence and uniqueness of the solution of Problem (P ) as well as (2.3). Next we prove (2.4). Let α ∈ (0, 1), p : We apply Lemma 2.3 and the embedding in Lemma 2.4 on domain Q 1 0 to obtain Similarly, we apply Lemma 2.3 and the embedding in Lemma 2.4 on the domains Q k+1 k and Q k+3/2 k+1/2 to obtain and a similar one on the domain Q k+3/2 k+1/2 . Finally, we deduce from the fact that k can be chosen arbitrarily large that which yields (2.4).

A version of Lojasiewicz inequality.
The main result of this section is the Lojasiewicz inequality stated in Theorem 3.7 below. More precisely, we prove a version of Lojasiewicz inequality for the functional E which coincides with the functional E on the solution orbits. We set This section is organized as follows: In Section 3.1, as a preparation for the proof of Theorem 3.7, we prove the differentiability of E and compute its derivative. The definition and some equivalent conditions of a critical point are given. The Lojasiewicz inequality is proved in Section 3.2.
3.1. Some preparations. We define the spaces Let V * be the dual space of V . We identify H with its dual to obtain: where the embeddings V → H, H → V * are continuous, dense and compact (see e.g. [19, p. 677]). We use ·, · to denote the duality product between V * and V . We denote by L(X, Y ) the space of bounded linear operators from a Banach space X to a second Banach space Y , and we write L(X) := L(X, X).
We also define the spaces equipped with the norm · L p (Ω) := · L p (Ω) and equipped with the norm · Xp := · W 2,p (Ω) . Throughout the sequel, we denote by C ≥ 0 a generic constant which may vary from line to line. We start with the following result.
be arbitrary and let g be a continuously differentiable function from IR to IR such that Proof. By Jensen's inequality and (3.3), Thus This completes the proof of Lemma 3.1.
The functional E is twice continuously Fréchet differentiable on V . We denote by E and L the first and second derivatives of E, respectively. Then (i) The first derivative As a consequence, Proof. We write E as the difference of E 1 and E 2 , where Obviously, E 1 is twice continuously Fréchet differentiable. Its derivatives are easily identified in the formulas (3.4) and (3.5). We now compute the first and second derivative of E 2 .
(i) By Taylor formula, It follows that We deduce from Lemma 3.1 that This implies that the first derivative E 2 exists and (ii) The Fréchet differentiability of E 2 is shown in a similar way. Choose p ∈ (2, +∞) such that V is continuously embedded in L p (Ω). Let T be the linear mapping from V to V * given by We will use below a generalized Hölder inequality based on the identity For every u, h, k ∈ V and for we have Consequently, we have which together with (3.9) follows that

SAMIRA BOUSSAÏD, DANIELLE HILHORST AND THANH NAM NGUYEN
Therefore, We also note that Hence which implies the continuity of E 2 . Finally, (3.6) is an immediate consequence of (3.5).
We define a continuous bilinear form from V × V → IR by The following lemma is an immediate consequence of the Lax-Milgram theorem (cf. [3, Corollary 5.8]). We omit its proof.
Corollary 2. The first and second derivatives of E can be represented in V * as: We also note that This together with (3.4) implies that Identity (3.12) may be proved in a similar way.
Lemma 3.4. Let L p (Ω), X p be the Banach spaces as in (3.1) and (3.2). Assume that p ≥ 2. Then, for any g ∈ L p (Ω), there exists a unique solution u ∈ X p of the equation Proof. It follows from Lemma 3.3 that the equation has a unique solution u ∈ V so that it is enough to prove that u ∈ X p . For this purpose, we consider the elliptic problem −∆ũ = g in Ω, ∂ νũ = 0 on ∂Ω.
Since g ∈ H, we apply the Fredholm alternative to deduce that this problem possesses a unique solutionũ ∈ V . Note that g ∈ L p (Ω), so that we deduce from [2] thatũ ∈ W 2,p (Ω) so that alsoũ ∈ X p . On the other hand, for all v ∈ V , we have Therefore,ũ coincides with the unique solution of equation (3.14). In other words, Then (3.13) follows.
Definition 3.5. We say that ϕ ∈ V is a critical point of E if Lemma 3.6. For every ϕ ∈ V , the following assertions are equivalent: (i) ϕ is a critical point of E, (ii) ϕ ∈ X 2 and ϕ satisfies the equations in Ω, Proof. (i) ⇒ (ii) Assume that ϕ ∈ V is a critical point of E. We deduce from (3.11) that Sincef (ii) ⇒ (i) It follows from (3.11) that which together with (3.13) implies that where the last identity follows from the fact that ϕ is a solution of Problem (S). Thus ϕ is a critical point of E.

15)
for all u − ϕ V ≤ σ. In this case, we say that E satisfies the Lojasiewicz inequality in ϕ. The number θ will be called the Lojasiewicz exponent.
We check below that all hypotheses in [8,Corollary 3.11] are satisfied so that the result of Theorem 3.7 will follow from [8, Corollary 3.11]. We need the following result. where codim Rg L(ϕ) := dim(V * / Rg L(ϕ)). As a consequence, V * is the direct sum of Rg L(ϕ) and ker L(ϕ).
Proof. We first prove that the linear operator Therefore T is continuous from V to H, which together with the compactness of the embedding H → V * implies that T is compact from V to V * .
Next, since A is an isomorphism from V onto V * , it is also a Fredholm operator of index ind A := dim ker A − codim Rg A = 0.
It follows that L(ϕ) = A + T , as a sum of a Fredholm operator and a compact operator, is also a Fredholm operator with the same index (cf. [3, p. 168]). This completes the proof of Lemma 3.8.
Before proving Theorem 3.7, we recall the definition of an analytic map on a neighborhood of a point (cf. [32,Definition 8.8,p. 362]). A map T from a Banach space X into a Banach space Y is called analytic on a neighborhood of z ∈ X if there exists ε > 0 such that for all h ∈ X, h X ≤ ε, Here, L i (X, Y ) is the space of bounded i-linear operators from X i to Y .
Proof of Theorem 3.7. In order prove Theorem 3.7, we apply [8,Corollary 3.11] for where p > N . In this case, there holds the embedding for all u ∈ X p . In view of Lemma 3.8, it is sufficient to prove that E is analytic in a neighborhood of ϕ. Indeed, let ε be small enough such that for all h ∈ X p with h Xp ≤ ε, we have we perform a Taylor's expansion to deduce for all h ∈ X p with h Xp ≤ ε that It follows that

SAMIRA BOUSSAÏD, DANIELLE HILHORST AND THANH NAM NGUYEN
We now prove that T i ∈ L i (X p , L p (Ω)). For all h 1 , . . . , h i ∈ X p , and 1 < i ≤ n, we have which implies that T i ∈ L i (X p , L p (Ω)) for all 1 < i ≤ n. In the case i = 1, since −∆ is linear, continuous from X p to L p (Ω), we easily deduce that T 1 ∈ L(X p , L p (Ω)). Therefore E is analytic on a neighborhood of ϕ. This completes the proof of Theorem 3.7.

Large time behavior.
Theorem 4.1. Let (H 0 ) hold and let u be the unique solution of Problem (P ).
Then there exists a function ϕ such that and ϕ is a solution of the stationary problem in Ω, This section is devoted to the proof of Theorem 4.1 by applying the Lojasiewicz inequality. In some case we also compute the limit stationary solution (see Theorem 4.5 below).
Proof. (i) In view of Corollary 1, for t > 0 we have As a consequence, for all 0 < s ≤ t < ∞ (ii) We recall that the functionF is bounded on IR. Therefore the function t → E(u(t)), which is nonincreasing and bounded from below, converges to a limit as t → ∞.  (ii) For all ϕ ∈ ω(u 0 ) E(ϕ) = e, where e is defined as in Lemma 4.2(ii). (iii) Let ϕ ∈ ω(u 0 ) then s 1 ≤ ϕ ≤ s 2 and is a stationary solution of Problem (P ), which implies that in Ω, Proof. (i) This is an immediate consequence of the relative compactness of solution orbits in H 1 (Ω) which is a consequence of Theorem 2.2.
We deduce from the continuity of E on H 1 (Ω) that where e is as in Lemma 4.2.
(iii) Since s 1 ≤ u(x, t) ≤ s 2 for all x ∈ Ω, t > 0, It follows that s 1 ≤ ϕ(x) ≤ s 2 for all x ∈ Ω. (4.2) Next, we prove that ϕ is a stationary solution. We denote here by u(t; w) the solution of Problem (P ) corresponding to initial function w. Let {t n } be such that u(t n ; u 0 ) → ϕ in H 1 (Ω) as n → ∞.
This implies in particular that u(t n ; u 0 ) → ϕ in L 2 (Ω) as n → ∞.
In view of Lemma 4.2 , we have t ≥ s > 0.
As a consequence, for all t > 0, u t (t; ϕ) = 0. In other words, u(t; ϕ) the solution of Problem (P ) with the initial function ϕ is independent of time. Therefore ϕ is a stationary solution of Problem (P ), which implies that in Ω, The identity follows from the mass conservation property.
Proof of Theorem 4.1. We will first prove Theorem 4.1 in the case Ω u 0 (x) = 0.
By the mass conservation property, we have  It follows from Lemma 4.4(iii) and Lemma 3.6 that for all ϕ ∈ ω(u 0 ), and that ϕ is a critical point of E. We apply Theorem 3.7 to deduce that E satisfies the Lojasiewicz inequality in the neighborhood of every ϕ ∈ ω(u 0 ). In other words, for every ϕ ∈ ω(u 0 ) there exist constants θ ∈ (0, 1 2 ], C ≥ 0 and δ > 0 such that Since E is continuous on V , we may choose δ small enough so that It follows from the compactness of ω(u 0 ) in V that there exists a neighborhood U of ω(u 0 ) composed of finitely many balls B j , j = 1, ..., J, with center ϕ j and radius δ j . In each of the ball B j , inequality (4.6) and the Lojasiewicz inequality (4.5) hold for some constants θ j and C j . We defineθ = min {θ j , j = 1, ..., J} and C = max {C j , j = 1, ..., J} to deduce from (4.4), (4.5) and (4.6) that It follows from Lemma 4.4(iv) that there exists t 0 ≥ 0 such that u(t) ∈ U for all t ≥ t 0 . Hence, for every t ≥ t 0 , there holds where we have also used (4.1). Note that for all t ≥ t 0 , E (u(t)) ∈ H and it can be written of the form Applying the continuous embedding H → V * , we have whereĈ is a positive constant. Combining (4.7) and (4.8) we obtain Here C =θ CĈ . Thus for all t 0 ≤ t 1 ≤ t 2 . Therefore u(t 1 ) − u(t 2 ) L 2 (Ω) tends to zero as t 1 → ∞ so that {u(t)} is a Cauchy sequence in H. As a consequence, there exists ϕ ∈ H such that lim t→∞ u(t) = ϕ exists in H, hence by the relative compactness of solution orbits in C 1 (Ω) we have lim In the general case, when instead of considering Problem (P ), we consider the Problem ( P ):  Then Proof. Let ϕ ∈ ω(u 0 ); it is sufficient to show that in Ω, Then we multiply the partial differential equation in (S) by ϕ and integrate over Ω to obtain Thus by Poincaré inequality Ω |ϕ − m 0 | 2 ≤ 0, which yields (4.9).
5. Convergence rate. In this section, we evaluate the rate of the convergence of the solution to the stationary solution. The proof is based once more on the Lojasiewicz inequality. We consider two cases: the Lojasiewicz exponent θ = 1 2 and θ ∈ (0, 1 2 ). These cases were studied by Haraux and Jendoubi [14] and Haraux, Jendoubi and Kavian [15]. 5.1. The case θ = 1 2 . We will apply the following result. Lemma 5.1 (see [14], Lemma 2.2). Let t 0 ≥ 0 be arbitrary. Assume that there exist positive constants γ and a such that Then for all τ ≥ t ≥ t 0 , Theorem 5.2. Let (H 0 ) hold. Assume further that Theorem 3.7 holds for θ = 1 2 ; then there exist positive constants K, δ such that u(t) − ϕ L ∞ (Ω) ≤ K exp(−δt) for all t ≥ 0.
Proof. As in the proof of Theorem 4.1, it is sufficient to prove this result for the function u with the assumption that where C 2 := 1/C 2 1 . We also note that is the unique solution of the differential equation Therefore, by [16, Theorem 6.1, page 31] and the differential inequality (5.2), we deduce that for all t ≥ T 0 E(u(t)) − E(ϕ) ≤ E(u(T 0 )) − E(ϕ) exp(−C 2 (t − T 0 )).

5.2.
The case θ ∈ (0, 1 2 ). We will apply the following lemma. Lemma 5.3 (see [15], Lemma 3.3). Let t 0 > 0 be arbitrary. Assume that there exist two positive constants α and K such that Proof. As in the proof of Theorem 4.1, it is sufficient to prove this result for the function u in the case