THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES RALPH CHILL Abstract. In this article we study the Lojasiewicz-Simon gradient inequality for energy functionals deﬁned on Hilbert spaces, using the so-called critical manifold. We indicate applications to partial diﬀerential equations.

1. Introduction Let V and H be real Hilbert spaces such that V is densely and continuously embedded into H (we write: V ֒→ H). We will identify H with its dual H ∗ so that V ֒→ H = H ∗ ֒→ V ∗ . Some important evolution equations can abstractly be written in the Hilbert space H or V ∗ in the form (1)

u(t) ˙ + E ′ (u(t)) = 0.

Here E : V → R is a differentiable functional. Finite dimensional gradient systems, semilinear diffusion equations and Cahn-Hilliard equations are some of the examples which can be written in this form. The function E is a strict Lyapunov function for (1) in the sense that if u ∈ C 1 (R+ ; V ) is a solution of (1), then the composition E(u) is nonincreasing, and if E(u) is constant then u is constant. In many applications, this property can even be verified for solutions which are only continuous with values in V and differentiable with values in H or V ∗ . By La Salle’s invariance principle, if u has relatively compact range with values in V , then the ω-limit set \ ω(u) = {u(s) : s ≥ t} t≥0

is non-empty, compact, connected and consists only of stationary solutions ϕ ∈ V , i.e. solutions of the stationary problem E ′ (ϕ) = 0, [16]. Date: May 1, 2006. 1

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An important question in the qualitative theory of (1) is the following: given a global solution with relatively compact range in V , does it converge to a single stationary solution? If the set of stationary solutions is discrete, then the answer is yes; this follows from La Salle’s invariance principle. In general, however, solutions with relatively compact range need not converge; there are counterexamples for finite dimensional gradient systems, [32], and for heat equations [33], [34]. By an idea due to Lojasiewicz one can prove covergence even if the set of stationary solutions is not discrete but a continuum, [27]. We will not repeat the convergence proof here, but we point out that it depends on the validity of the Lojasiewicz inequality which in the following we will call Lojasiewicz-Simon inequality for L. Simon extended Lojasiewicz’ ideas to the infinite dimensional case, [35]. Starting with the work of Jendoubi in [25], [24] the LojasiewiczSimon inequality has in the last decade been successfully applied in order prove convergence results for solutions of a variety of infinite dimensional gradient or gradient-like systems, such as Cahn-Hilliard equations [21], [7], degenerate diffusion equations [15], [8], second order ODEs [17], [3], damped wave equations [18], evolutionary integral equations [1], [2], [6], or non-autonomous equations [23], [9], [10]. These applications are in fact our motivation for studying the Lojasiewicz-Simon inequality. In this note, we will explain the proof of the Lojasiewicz-Simon inequality in the infinite dimensional Hilbert space case, we discuss the abstract results and we indicate some applications to partial differential equations. 2. Main results We assume that E ∈ C 2 (V ). Then E ′ ∈ C 1 (V ; V ∗ ), where V ∗ is the dual of V . The second derivative E ′′ will be denoted by L. Note that L(u) may be identified with a bounded linear operator V → V ∗ or with a bounded bilinear form V × V → R. We will adopt the first identification as a bounded linear operator in the following. Let ϕ ∈ V be a stationary solution, i.e. E ′ (ϕ) = 0. We say that E satisfies the Lojasiewicz-Simon inequality near ϕ if there exists a neighbourhood U ⊂ V of ϕ and constants θ ∈ (0, 21 ], C ≥ 0 such that (2)

|E(u) − E(ϕ)|1−θ ≤ C kE ′ (u)kV ∗ for every u ∈ U.

The constant θ will be called the Lojasiewicz exponent. Although the formulation of the Lojasiewicz-Simon inequality requires only E ∈ C 1 (V ), one way of proving it in infinite dimensions requires E ∈ C 2 (V ). In fact, the operator L(ϕ) plays an important role.

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We will assume that L(ϕ) is a Fredholm operator, i.e. the kernel ker L(ϕ) = {u ∈ V : L(ϕ)u = 0} is finite dimensional and the range rg L(ϕ) = {L(ϕ)u : u ∈ V } is closed in V ∗ and has finite codimension. The condition on the codimension of the range is automatic in our situation since L(ϕ) is symmetric by Schwarz’ theorem, [4, Th´eor`eme 5.1.1]. Let P : H → H be the orthogonal projection onto ker L(ϕ). Clearly, P is also a bounded projection in V , and by symmetry (P = P ∗ in H) P extends to a bounded projection in V ∗ . In each space, the range of P is ker L(ϕ). Lemma 1. The set S := {u ∈ V : (I − P )E ′(u) = 0} is locally near ϕ a differentiable manifold satisfying dim S = dim ker L(ϕ). If E ∈ C k (V ) for some k ≥ 2, then S is a C k−1 -manifold. If E is analytic, then S is analytic. Proof. Since P is a bounded linear projection in V , the space V is the direct topological sum of the two subspaces V0 = ker L(ϕ) = rg P and Let

V1∗

V1 = ker P. be the kernel of P in V . Note that V1∗ is the dual of V1 . More precisely, ∗

V1 ֒→ H1 = H1∗ ֒→ V1∗ , where H1 is the kernel of P in H. Consider the function G : V = V0 ⊕ V1 → V1∗ , u = u0 + u1 7→ (I − P )E ′ (u). Since E ′ is C 1 , the function G is C 1 . Moreover, G′ (ϕ) = (I − P )L(ϕ). ∂G (ϕ) = (I − P )L(ϕ)|V1 : V1 → V1∗ is We claim that the partial derivative ∂v 1 boundedly invertible. In fact, this follows from the spectral theory of symmetric operators. ∂G We first prove that ∂v (ϕ) is injective. Let u ∈ V1 be such that 1 (I − P )L(ϕ)u = 0.

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Then L(ϕ)u ∈ rg P . On the other hand, one always has L(ϕ)u ∈ ker P . Recall that the operator L(ϕ) is symmetric by Schwarz’ theorem, [4, Th´eor`eme 5.1.1]. This and the symmetry of P imply that for every v ∈ V , hP L(ϕ)u, viV ∗ ,V

= hL(ϕ)u, P viV ∗ ,V = hL(ϕ)P v, uiV ∗ ,V = 0,

where in the last inequality we have used that P projects onto the kernel of L(ϕ). Hence, L(ϕ)u ∈ ker P . Together with L(ϕ)u ∈ rg P this implies L(ϕ)u = 0 or u ∈ ker L(ϕ). However, since u was supposed to be in V1 , the complement of ker L(ϕ), this implies u = 0. ∂G We next prove that ∂v (ϕ) is surjective. Using again the symmetry of P and 1 ∂G ∂G (ϕ) is symmetric. Since ∂v (ϕ) is injective, this of L(ϕ), it is easy to see that ∂v 1 1 operator must have dense range by the Hahn-Banach theorem. By assumption, ∂G L(ϕ) has closed range, and hence the operator ∂v (ϕ) has closed range. This 1 proves surjectivity. ∂G (ϕ) is boundedly invertible. By the By the bounded inverse theorem, ∂v 1 implicit function theorem [4, Th´eor`eme 4.7.1], there exists a neighbourhood U0 ⊂ V0 of ϕ0 , a neighbourhood U1 ⊂ V1 of ϕ1 and a function g ∈ C 1 (U0 ; U1 ) such that g(ϕ0 ) = ϕ1 and {u ∈ U = U0 + U1 : G(u) = 0} = {(u0 , g(u0)) : u0 ∈ U0 }. By definition of the function G and the set S, the set on the left-hand side of this equality is just the intersection of S with the neighbourhood U = U0 + U1 of ϕ in V . Hence we have proved the first claim. The second claim, i.e. higher regularity of the manifold S in the case of higher regularity of E, follows immediately from the implicit function theorem. We call the set S from Lemma 1 critical manifold. It may not be a submanifold of V , but by Lemma 1 it is locally near ϕ a submanifold. For our purposes this is sufficient since the Lojasiewicz-Simon inequality is only a local property, too. Theorem 2. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold as in Lemma 1. Assume that the restriction E|S satisfies the Lojasiewicz-Simon inequality near ϕ, i.e. there exists a neighbourhood U ⊂ V of ϕ and constants θ ∈ (0, 21 ], C ≥ 0 such that |E(u) − E(ϕ)|1−θ ≤ C kE ′ (u)kV ∗ for every u ∈ U ∩ S.

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Then E itself satisfies the Lojasiewicz-Simon inequality near ϕ for the same Lojasiewicz exponent θ. Proof. Choose the neighbourhood U = U0 + U1 ⊂ V of ϕ and the implicit function g : U0 → U1 as in the proof of Lemma 1. Suppose that U is sufficiently small so that the restriction E|S satisfies the Lojasiewicz-Simon inequality in U ∩ S. We define a nonlinear projection Q : U → U onto S by Qu = Q(u0 + u1 ) := u0 + g(u0). Note that Qu really belongs to S and that u − Qu belongs to the space V1 . For every u ∈ U Taylor’s theorem implies E(u) − E(Qu) = 1 = hE ′ (Qu), u − QuiV ∗ ,V + hL(Qu)(u − Qu), u − QuiV ∗ ,V + o(ku − Quk2 ). 2 By definition of V1 , and by definition of the manifold S, hE ′ (Qu), u − QuiV ∗ ,V

= hE ′ (Qu), (I − P )(u − Qu)iV ∗ ,V = h(I − P )E ′ (Qu), u − QuiV ∗ ,V = 0,

i.e. the first term on the right-hand side of the Taylor expansion of E is zero. Moreover, if we choose the neighbourhood U small enough, then L is uniformly bounded on U by continuity and therefore (3)

|E(u) − E(Qu)| ≤ C ku − Quk2V .

From now on, the constant C may vary from line to line. By the definition of differentiability, (4)

E ′ (u) − E ′ (Qu) = L(Qu)(u − Qu) + o(ku − Quk).

We apply the projection I − P to this equality and use the definition of S in order to obtain (I − P )E ′ (u) = (I − P )L(Qu)(u − Qu) + o(ku − Quk). By the proof of Lemma 1, the operator (I − P )L(ϕ) : V1 → V1∗ is boundedly invertible. Hence, by continuity and if we choose U small enough, then (I − P )L(Qu) : V1 → V1∗ is boundedly invertible for all u ∈ U and the inverses are uniformly bounded in U. As a consequence, there exists a constant C ≥ 0 such that for every u ∈ U (5)

ku − QukV ≤ C k(I − P )E ′ (u)kV ∗ ≤ C kE ′ (u)kV ∗ .

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Combining the estimates (3) and (5) with the assumption that E|S satisfies the Lojasiewicz-Simon inequality in U ∩ S, we obtain that for every u ∈ U |E(u) − E(ϕ)| ≤ |E(u) − E(Qu)| + |E(Qu) − E(ϕ)| 1/(1−θ)

≤ C kE ′ (u)k2V ∗ + C kE ′ (Qu)kV ∗

.

By (4) and (5), kE ′ (Qu)kV ∗ ≤ kE ′ (u)kV ∗ + C ku − QukV ≤ C kE ′ (u)kV ∗ . As a consequence, for every u ∈ U, 1/(1−θ)

|E(u) − E(ϕ)| ≤ C kE ′ (u)k2V ∗ + kE ′ (u)kV ∗

.

Choosing U sufficiently small, we can by continuity assume that kE ′ (u)kV ∗ ≤ 1 for every u ∈ U. Since θ ∈ (0, 21 ], we then obtain 1/(1−θ)

|E(u) − E(ϕ)| ≤ C kE ′ (u)kV ∗

for every u ∈ U,

and this is the claim.

There are basically two important cases in which the assumption on E from Theorem 2 can be verified. The first one uses Lojasiewicz’ gradient inequality for real analytic functions on Rn , [28]. The second one is more elementary. Applications of both corollaries will be illustrated by our examples in the last section. Corollary 3. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold S as in Lemma 1. If S is analytic and if the restriction E|S is analytic, then E satisfies the Lojasiewicz-Simon inequality near ϕ. Proof. By assumption on the linearization L(ϕ) and by Lemma 1, S is finite dimensional. By Lojasiewicz’ classical result [28], the restriction E|S satisfies the Lojasiewicz-Simon inequality near ϕ. The claim follows from Theorem 2. Corollary 4. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold S as in Lemma 1. Assume that the set of stationary solutions S0 = {u ∈ V : E ′ (u) = 0} forms a neighbourhood of ϕ in S. Then E satisfies the Lojasiewicz-Simon inequality for the Lojasiewicz exponent θ = 21 .

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Proof. By assumption, the derivative E ′ is constant zero in a neighbourhood of ϕ in S. This implies that the restriction E|S is constant in the same neighbourhood. A constant function trivially satisfies the Lojasiewicz-Simon inequality for the Lojasiewicz exponent θ = 12 . The claim follows from Theorem 2.

3. Remarks Remark 5. The formulation and the proof of the Lojasiewicz-Simon inequality are independent of the Hilbert space H. In fact, the Hilbert space only allows us to identify the dual space V ∗ and thus also the derivative E ′ . Different choices of the Hilbert space H and different choices of the inner product in H lead to different identifications of the dual space V ∗ and the derivative E ′ ; see the examples in the next section. The value kE ′ (u)kV ∗ , however, does not change. The validity of the Lojasiewicz-Simon inequality is thus independent of the choice of H. Also the proof of the inequality can be formulated without using the Hilbert space H; simply identify V ∗ with V , or see [5]. In our proof the space H was only used for choosing a particular projection P . The Gelfand triple V ֒→ H = H ∗ ֒→ V ∗ is solely motivated by applications to partial differential equations. Remark 6. The critical manifold is in general not unique. The definition of the critical manifold S depends on the choice of the projection P . In order to simplify the presentation of the proof we have chosen the orthogonal projection (orthogonal in H) onto the kernel of the linearization L(ϕ). However, as remarked above, the space H and the inner product in H are not unique and thus also the projection P is not unique. Therefore, in general the critical manifold can vary when varying the projection P . On the other hand, the critical manifold always contains the set of stationary solutions. If the critical manifold coincides with the set of stationary solutions, then it is of course independent of the choice of the projection P . Remark 7. The critical manifold is not the center manifold. This follows already from the non-uniqueness of the critical manifold. However, both manifolds are tangent to the kernel of L(ϕ), and both manifolds have the same dimension than the kernel of L(ϕ). Both manifolds are critical in the following senses: in the case of existence of a center manifold, the difficult dynamics (different from exponential stability or exponential expansiveness) take place there. And by Theorem 2, the critical manifold is the set where the Lojasiewicz-Simon inequality may fail, but if the inequality is satisfied in the critical manifold then it is satisfied everywhere.

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Note, however, that if the critical manifold consists only of stationary solutions (at least locally) then the critical manifold and the center manifold do coincide and the dynamics are trivial in the center manifold. Remark 8. The Lojasiewicz-Simon inequality can be refined. Given E ∈ C 1 (V ) and ϕ ∈ V a stationary solution, we say that E satisfies the (generalized) Lojasiewicz-Simon inequality if there exists a neighbourhood U ⊂ V of ϕ and a strictly increasing function θ ∈ C([0, ∞[) ∩ C 1 (]0, ∞[) such that θ(0) = 0 and 1 ≤ kE ′ (u)kV ∗ . ′ θ (|E(u) − E(ϕ)|) This definition is due to Haraux & Jendoubi [20] and Huang [22]. The classical Lojasiewicz-Simon inequality is obtained by taking θ(t) = ctθ . Considering this type of inequality can be an advantage because of two reasons. First, there are more functionals which may satisfy the generalized Lojasiewicz-Simon inequality since we are allowing functions θ which are very slowly increasing. The proof of convergence to stationary solutions works also for this type of inequality. Second, it is known that the decay rate to stationary solutions depends on the function θ (or before: the Lojasiewicz exponent). Thus, also functions θ which are between two powers, like e.g. the function θ(t) = tα log(e + 1t ), are of interest. 4. Applications The applications which we describe in this section are all based on the same functional E! By this we want to show that the proof of the Lojasiewicz-Simon inequality is a problem which is independent of particular evolution equations. Let Ω ⊂ Rn be an open set and let f ∈ C 2 (Ω × R; R) be subcritical in the sense that ∂2f | 2 (x, s)| ≤ C|s|p for every s ∈ R, ∂s for some constant C ≥ 0 and some p ≥ 1 satisfying ( 4 if n ≥ 3, n−2 p< +∞ if n = 2. If n = 1, then we impose no growth condition on f . If Ω is bounded then the 2 growth condition on ∂∂sf2 for small s ∈ R can be relaxed to mere boundedness. Define the functional E : H01 (Ω) → R by Z Z 1 2 E(u) := |∇u| + F (x, u), 2 Ω Ω

THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES

where F (x, u) =

Ru 0

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f (x, s) ds. It is an exercise to show that E is C 2 .

The functional E plays an important role in several applications. We mention some examples of gradient systems, i.e. systems which can abstractly be written in the form (1). Throughout these examples we put V = H01 (Ω). Example 9. Let H = L2 (Ω), equipped with the usual inner product. Then V embeds continuously and densely into H. When we identify H = L2 (Ω) with its dual, then V ∗ is the distribution space H −1(Ω) and E ′ (u) = −∆u + f (x, u). The evolution equation (1) thus becomes the semilinear heat equation ut − ∆u + f (x, u) = 0, subject to Dirichlet boundary conditions. The asymptotic behaviour of this semilinear heat equation has been studied in many articles. If Ω ⊂ R is a bounded interval, then every solution with relatively compact range in H01 converges to a stationary solution, [36], [29]. Zelenyak uses in his proof in [36] in principle the same idea than Lojasiewicz, while Matano’s proof in [29] is completely different. Zelenyak’s proof contains in addition a nice argument by contradiction the idea of which is as follows: if the solution does not converge then the ω-limit set is a connected continuum in the critical manifold. The critical manifold, however, is one dimensional by Lemma 1 and classical theory of ordinary differential equations. We thus find a point in the ω-limit set which is in the interior of the ω-limit set with respect to the critical manifold S. Near this point, the functional satisfies the LojasiewiczSimon inequality by Corollary 4. However, if the Lojasiewicz-Simon inequality holds near some point of the ω-limit set, then the solution must converge; a contradiction! This argument by contradiction has been extended by Haraux & Jendoubi in [19] and has been applied in many other situations including wave equations. If Ω ⊂ Rn is a bounded set and if f is analytic in the second variable, uniformly with respect to the first variable, then one may apply Corollary 3 in order to see that E satisfies the Lojasiewicz-Simon inequality near every stationary solution (and hence every bounded solution of the heat equation converges). Note, however, that in general E is not analytic on the energy space V = H01 (Ω) so that analyticity of the critical manifold S and of the restriction E|S do not follow from Lemma 1 directly. However, E is analytic on H01 (Ω) ∩ L∞ (Ω), and analyticity of S and of E|S is then deduced from elliptic regularity and arguments similar to those in the proof of Lemma 1; see e.g. [35], [25], [18].

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It is known that without any further conditions on f , solutions with relatively compact range in H01 (Ω) need not converge, [33], [34]. These counterexamples show that the Lojasiewicz-Simon inequality is in general not satisfied by the functional E. In the cases described above (i.e. when Ω is bounded) the Fredholm property of L(ϕ) follows from the Rellich-Kondrachov theorem and the spectral theory of compact operators. However, the Fredholm property can also be proved in special situations by using the spectral theory of Schr¨odinger operators. For example, if Ω = Rn , if f (x, s) = s − |s|p−1s for some p < n+2 , and if n−2 ϕ 6= 0 is a positive stationary solution (in fact, there exists only one up to translations), then L(ϕ) is Fredholm. Moreover, dim ker L(ϕ) = n and this is exactly the dimension of the set of positive stationary solutions which are obtained by translations of ϕ. Hence, the critical manifold (near a positive stationary solution) consists only of stationary solutions, and therefore the Lojasiewicz-Simon inequality holds by Corollary 4. These facts have been proved and used by Feireisl & Petzeltov´a [14] and Faˇsangov´a [11] in order to show that positive bounded solutions of the heat equation converge. See also [10] for a nonautonomous heat equation. Positivity is in general necessary for convergence as was shown by Faˇsangov´a & Feireisl [12]. Example 10. Let H = L2 (Ω), equipped with the inner product Z 1 (u, v)m := uv , m Ω

where m ∈ L∞ (Ω) is a positive function satisfying m1 ∈ L∞ (Ω). This inner product is clearly equivalent to the usual inner product. Still, the space V is dense in H and V ∗ = H −1 (Ω). However, due to the change of the inner product, the derivative E ′ becomes E ′ (u) = −m∆u + mf (x, u), and when we put g = mf , then the evolution equation (1) becomes the semilinear diffusion equation ut − m∆u + g(x, u) = 0. Example 11. Assume that Ω is bounded and let H = H01 (Ω), equipped R with the inner product (u, v)H01 = Ω ∇u∇v. This means in particular that we identify V with its dual V ∗ . With this identification, the derivative E ′ becomes E ′ (u) = u + (−∆)−1 f (x, u). The resulting gradient system is of particular interest in numerical computations. Here one is interested in solving the elliptic problem −∆u + f (x, u) = 0,

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i.e. in finding the stationary solutions of the functional E. Of course, these do not change with different identifications of V ∗ . The resulting evolution equation for the so-called Sobolev gradient of E above is a steepest descent method. Clearly, the gradient system obtained for this choice of H is in numerical computations approximated by a finite dimensional gradient system, but the advantage to approximate just this system and not for example the heat equation lies in the better regularity of the Sobolev gradient resulting in better convergence behaviour of the steepest descent method. For the theory of Sobolev gradients and the applications to numerics we refer to Neuberger [30], [31]. Example 12. Assume again that Ω is bounded, equip H01 (Ω) with the inner product from the preceeding example, and let H = H −1 (Ω), equipped with the inner product (u, v)H −1 = ((−∆)−1 u, (−∆)−1 v)H01 , where (−∆)−1 is the inverse of the negative Dirichlet-Laplace operator. In this example we have V ∗ = H −3 (Ω) and the derivative E ′ becomes E ′ (u) = ∆(∆u − f (x, u)), subject to the Dirichlet boundary conditions u|∂Ω = ∆u|∂Ω = 0. With this identification the evolution equation (1) results into the Cahn-Hillard equation ut + ∆(∆u − f (x, u)) = 0, subject to the above Dirichlet boundary conditions. It should be noted that these Dirichlet boundary conditions are not the usual ones for the CahnHilliard equation and thus this example is artificial. On the other hand, the more usual Neumann boundary conditions are obtained by simply considering the energy space V = H 1 (Ω) (or V the space of H 1 functions with zero mean) instead of H01 (Ω). Concerning the Lojasiewicz-Simon inequality everything then works similarly provided the set Ω is sufficiently regular so that H 1 (Ω) embeds compactly into L2 (Ω) (this ensures the Fredholm property of the linearization L(ϕ)). Convergence of bounded solutions of the Cahn-Hillard equation has been proved by Hoffmann & Rybka [21]. For the Cahn-Hillard equation with dynamic boundary conditions, see [7]. Example 13. We finally discuss an evolution equation which is not a gradient system. Take again H = L2 (Ω) as in Example 9 so that E ′ (u) = −∆u+f (x, u). The functional E plays an important role also for the semilinear damped wave equation utt + ut − ∆u + f (x, u) = 0, subject to Dirichlet boundary conditions. Jendoubi was the first to apply the Lojasiewicz-Simon inequality to prove convergence to stationary solutions in this gradient-like system, [24]; see also [18]. The convergence results are similar

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to those for the semilinear heat equation since Jendoubi’s proof of convergence mainly uses the Lojasiewicz-Simon inequality for the functional E. Therefore, if Ω ⊂ R is a bounded interval, then every weak solution which is bounded in H01 converges to a stationary solution. The same holds true if Ω ⊂ Rn is bounded and if f is analytic in the second variable, uniformly with respect to the first one. In general, bounded solutions need not converge, [26]. For a convergence result for wave equations on the whole space Rn , see [13]. References 1. S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-ﬁeld model with memory, J. Evolution Equations 1 (2001), 69–84. 2. S. Aizicovici and H. Petzeltov´a, Asymptotic behavior of solutions of a conserved phaseﬁeld system with memory, J. Integral Equations Appl. 15 (2003), 217–240. 3. H. Attouch, X. Goudou, and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: global exploration of the local minima of a real-valued function asymptotic by analysis of a dissipative dynamical system, Commun. Contemp. Math. 2 (2000), 1–34. 4. H. Cartan, Calcul diﬀ´erentiel, Hermann, Paris, 1967. 5. R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), 572–601. 6. R. Chill and E. Faˇsangov´a, Convergence to steady states of solutions of semilinear evolutionary integral equations, Calc. Var. Partial Diﬀerential Equations 22 (2005), 321–342. 7. R. Chill, E. Faˇsangov´a, and J. Pr¨ uss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr. (2006), to appear. 8. R. Chill and A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Diﬀerential Equations (2006), to appear. 9. R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Analysis, Ser. A: Theory Methods 53 (2003), 1017–1039. 10. , Convergence to steady states of solutions of non-autonomous heat equations in RN , J. Dynam. Diﬀerential Equations (2006), to appear. 11. E. Faˇsangov´a, Asymptotic analysis for a nonlinear parabolic equation on R, Comment Math. Univ. Carolin. 39 (1998), 525–544. 12. E. Faˇsangov´a and E. Feireisl, The long-time behaviour of solutions to parabolic problems on unbounded intervals: the inﬂuence of boundary conditions, Proc. Roy. Soc. Edinburgh 129A (1999), 319–329. 13. E. Feireisl, Long-time behavior and convergence for semilinear wave equations on RN , J. Dynam. Diﬀerential Equations 9 (1997), 133–155. 14. E. Feireisl and H. Petzeltov´a, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Diﬀerential Integral Equations 10 (1997), 181–196. 15. E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Diﬀerential Equations 12 (2000), 647–673. 16. A. Haraux, Syst`emes dynamiques dissipatifs et applications, Masson, Paris, 1990. 17. A. Haraux and M. A. Jendoubi, Convergence of solutions to second-order gradient-like systems with analytic nonlinearities, J. Diﬀerential Equations 144 (1998), 313–320.

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18. 19. 20. 21. 22.

23. 24.

25. 26.

27.

28. 29. 30. 31. 32. 33. 34. 35. 36.

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, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Diﬀerential Equations 9 (1999), 95–124. , On the convergence of global and bounded solutions of some evolution equations, Pr´epublication du Laboratoire Jacques-Louis Lions R02003 (2002). , Oral communication, 2003. K.-H. Hoﬀmann and P. Rybka, Convergence of solutions to Cahn-Hillard equation, Commun. Partial Diﬀerential Equations 24 (1999), 1055–1077. S.-Z. Huang, Gradient Inequalities: with Applications to Asymptotic Behaviour and Stability of Gradient-like Systems, Mathematical Surveys and Monographs, vol. 126, Amer. Math. Soc., Providence, R.I., 2006. S.-Z. Huang and P. Tak´ aˇc, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., Ser. A, Theory Methods 46 (2001), 675–698. M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Diﬀerential Equations 144 (1998), 302– 312. , A simple uniﬁed approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998), 187–202. M. A. Jendoubi and P. Pol´ aˇcik, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Royal Soc. Edinburgh Sect A. 133 (2003), 1137– 1153. S. Lojasiewicz, Une propri´et´e topologique des sous-ensembles analytiques r´eels, Colloques internationaux du C.N.R.S.: Les ´equations aux d´eriv´ees partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, pp. 87–89. , Ensembles semi-analytiques, Preprint, I.H.E.S. Bures-sur-Yvette, 1965. H. Matano, Convergence of solutions of one-dimensional semilinear heat equations, J. Math. Kyoto Univ. 18 (1978), 221–227. J. W. Neuberger, Sobolev Gradients and Diﬀerential Equations, Lect. Notes Math., vol. 1670, Springer Verlag, Berlin, Heidelberg, New York, 1997. , Prospects for a central theory of partial diﬀerential equations, Math. Intelligencer 27 (2005), 47–55. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer Verlag, New York, Heidelberg, Berlin, 1982. P. Pol´ aˇcik and K. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Diﬀerential Equations 124 (1996), 472–494. P. Pol´ aˇcik and F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Diﬀerential Equations 186 (2002), 586–610. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983), 525–571. T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Diﬀer. Eq. 4 (1968), 17–22, Transl. from Diﬀer. Uravn. 4 (1968), 34–45.

Universit´ e Paul Verlaine - Metz, Laboratoire de Math´ ematiques et Apˆt. A, Ile du Saulcy, 57045 Metz plications de Metz et CNRS, UMR 7122, Ba Cedex 1, France E-mail address: [email protected]

1. Introduction Let V and H be real Hilbert spaces such that V is densely and continuously embedded into H (we write: V ֒→ H). We will identify H with its dual H ∗ so that V ֒→ H = H ∗ ֒→ V ∗ . Some important evolution equations can abstractly be written in the Hilbert space H or V ∗ in the form (1)

u(t) ˙ + E ′ (u(t)) = 0.

Here E : V → R is a differentiable functional. Finite dimensional gradient systems, semilinear diffusion equations and Cahn-Hilliard equations are some of the examples which can be written in this form. The function E is a strict Lyapunov function for (1) in the sense that if u ∈ C 1 (R+ ; V ) is a solution of (1), then the composition E(u) is nonincreasing, and if E(u) is constant then u is constant. In many applications, this property can even be verified for solutions which are only continuous with values in V and differentiable with values in H or V ∗ . By La Salle’s invariance principle, if u has relatively compact range with values in V , then the ω-limit set \ ω(u) = {u(s) : s ≥ t} t≥0

is non-empty, compact, connected and consists only of stationary solutions ϕ ∈ V , i.e. solutions of the stationary problem E ′ (ϕ) = 0, [16]. Date: May 1, 2006. 1

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An important question in the qualitative theory of (1) is the following: given a global solution with relatively compact range in V , does it converge to a single stationary solution? If the set of stationary solutions is discrete, then the answer is yes; this follows from La Salle’s invariance principle. In general, however, solutions with relatively compact range need not converge; there are counterexamples for finite dimensional gradient systems, [32], and for heat equations [33], [34]. By an idea due to Lojasiewicz one can prove covergence even if the set of stationary solutions is not discrete but a continuum, [27]. We will not repeat the convergence proof here, but we point out that it depends on the validity of the Lojasiewicz inequality which in the following we will call Lojasiewicz-Simon inequality for L. Simon extended Lojasiewicz’ ideas to the infinite dimensional case, [35]. Starting with the work of Jendoubi in [25], [24] the LojasiewiczSimon inequality has in the last decade been successfully applied in order prove convergence results for solutions of a variety of infinite dimensional gradient or gradient-like systems, such as Cahn-Hilliard equations [21], [7], degenerate diffusion equations [15], [8], second order ODEs [17], [3], damped wave equations [18], evolutionary integral equations [1], [2], [6], or non-autonomous equations [23], [9], [10]. These applications are in fact our motivation for studying the Lojasiewicz-Simon inequality. In this note, we will explain the proof of the Lojasiewicz-Simon inequality in the infinite dimensional Hilbert space case, we discuss the abstract results and we indicate some applications to partial differential equations. 2. Main results We assume that E ∈ C 2 (V ). Then E ′ ∈ C 1 (V ; V ∗ ), where V ∗ is the dual of V . The second derivative E ′′ will be denoted by L. Note that L(u) may be identified with a bounded linear operator V → V ∗ or with a bounded bilinear form V × V → R. We will adopt the first identification as a bounded linear operator in the following. Let ϕ ∈ V be a stationary solution, i.e. E ′ (ϕ) = 0. We say that E satisfies the Lojasiewicz-Simon inequality near ϕ if there exists a neighbourhood U ⊂ V of ϕ and constants θ ∈ (0, 21 ], C ≥ 0 such that (2)

|E(u) − E(ϕ)|1−θ ≤ C kE ′ (u)kV ∗ for every u ∈ U.

The constant θ will be called the Lojasiewicz exponent. Although the formulation of the Lojasiewicz-Simon inequality requires only E ∈ C 1 (V ), one way of proving it in infinite dimensions requires E ∈ C 2 (V ). In fact, the operator L(ϕ) plays an important role.

THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES

3

We will assume that L(ϕ) is a Fredholm operator, i.e. the kernel ker L(ϕ) = {u ∈ V : L(ϕ)u = 0} is finite dimensional and the range rg L(ϕ) = {L(ϕ)u : u ∈ V } is closed in V ∗ and has finite codimension. The condition on the codimension of the range is automatic in our situation since L(ϕ) is symmetric by Schwarz’ theorem, [4, Th´eor`eme 5.1.1]. Let P : H → H be the orthogonal projection onto ker L(ϕ). Clearly, P is also a bounded projection in V , and by symmetry (P = P ∗ in H) P extends to a bounded projection in V ∗ . In each space, the range of P is ker L(ϕ). Lemma 1. The set S := {u ∈ V : (I − P )E ′(u) = 0} is locally near ϕ a differentiable manifold satisfying dim S = dim ker L(ϕ). If E ∈ C k (V ) for some k ≥ 2, then S is a C k−1 -manifold. If E is analytic, then S is analytic. Proof. Since P is a bounded linear projection in V , the space V is the direct topological sum of the two subspaces V0 = ker L(ϕ) = rg P and Let

V1∗

V1 = ker P. be the kernel of P in V . Note that V1∗ is the dual of V1 . More precisely, ∗

V1 ֒→ H1 = H1∗ ֒→ V1∗ , where H1 is the kernel of P in H. Consider the function G : V = V0 ⊕ V1 → V1∗ , u = u0 + u1 7→ (I − P )E ′ (u). Since E ′ is C 1 , the function G is C 1 . Moreover, G′ (ϕ) = (I − P )L(ϕ). ∂G (ϕ) = (I − P )L(ϕ)|V1 : V1 → V1∗ is We claim that the partial derivative ∂v 1 boundedly invertible. In fact, this follows from the spectral theory of symmetric operators. ∂G We first prove that ∂v (ϕ) is injective. Let u ∈ V1 be such that 1 (I − P )L(ϕ)u = 0.

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Then L(ϕ)u ∈ rg P . On the other hand, one always has L(ϕ)u ∈ ker P . Recall that the operator L(ϕ) is symmetric by Schwarz’ theorem, [4, Th´eor`eme 5.1.1]. This and the symmetry of P imply that for every v ∈ V , hP L(ϕ)u, viV ∗ ,V

= hL(ϕ)u, P viV ∗ ,V = hL(ϕ)P v, uiV ∗ ,V = 0,

where in the last inequality we have used that P projects onto the kernel of L(ϕ). Hence, L(ϕ)u ∈ ker P . Together with L(ϕ)u ∈ rg P this implies L(ϕ)u = 0 or u ∈ ker L(ϕ). However, since u was supposed to be in V1 , the complement of ker L(ϕ), this implies u = 0. ∂G We next prove that ∂v (ϕ) is surjective. Using again the symmetry of P and 1 ∂G ∂G (ϕ) is symmetric. Since ∂v (ϕ) is injective, this of L(ϕ), it is easy to see that ∂v 1 1 operator must have dense range by the Hahn-Banach theorem. By assumption, ∂G L(ϕ) has closed range, and hence the operator ∂v (ϕ) has closed range. This 1 proves surjectivity. ∂G (ϕ) is boundedly invertible. By the By the bounded inverse theorem, ∂v 1 implicit function theorem [4, Th´eor`eme 4.7.1], there exists a neighbourhood U0 ⊂ V0 of ϕ0 , a neighbourhood U1 ⊂ V1 of ϕ1 and a function g ∈ C 1 (U0 ; U1 ) such that g(ϕ0 ) = ϕ1 and {u ∈ U = U0 + U1 : G(u) = 0} = {(u0 , g(u0)) : u0 ∈ U0 }. By definition of the function G and the set S, the set on the left-hand side of this equality is just the intersection of S with the neighbourhood U = U0 + U1 of ϕ in V . Hence we have proved the first claim. The second claim, i.e. higher regularity of the manifold S in the case of higher regularity of E, follows immediately from the implicit function theorem. We call the set S from Lemma 1 critical manifold. It may not be a submanifold of V , but by Lemma 1 it is locally near ϕ a submanifold. For our purposes this is sufficient since the Lojasiewicz-Simon inequality is only a local property, too. Theorem 2. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold as in Lemma 1. Assume that the restriction E|S satisfies the Lojasiewicz-Simon inequality near ϕ, i.e. there exists a neighbourhood U ⊂ V of ϕ and constants θ ∈ (0, 21 ], C ≥ 0 such that |E(u) − E(ϕ)|1−θ ≤ C kE ′ (u)kV ∗ for every u ∈ U ∩ S.

THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES

5

Then E itself satisfies the Lojasiewicz-Simon inequality near ϕ for the same Lojasiewicz exponent θ. Proof. Choose the neighbourhood U = U0 + U1 ⊂ V of ϕ and the implicit function g : U0 → U1 as in the proof of Lemma 1. Suppose that U is sufficiently small so that the restriction E|S satisfies the Lojasiewicz-Simon inequality in U ∩ S. We define a nonlinear projection Q : U → U onto S by Qu = Q(u0 + u1 ) := u0 + g(u0). Note that Qu really belongs to S and that u − Qu belongs to the space V1 . For every u ∈ U Taylor’s theorem implies E(u) − E(Qu) = 1 = hE ′ (Qu), u − QuiV ∗ ,V + hL(Qu)(u − Qu), u − QuiV ∗ ,V + o(ku − Quk2 ). 2 By definition of V1 , and by definition of the manifold S, hE ′ (Qu), u − QuiV ∗ ,V

= hE ′ (Qu), (I − P )(u − Qu)iV ∗ ,V = h(I − P )E ′ (Qu), u − QuiV ∗ ,V = 0,

i.e. the first term on the right-hand side of the Taylor expansion of E is zero. Moreover, if we choose the neighbourhood U small enough, then L is uniformly bounded on U by continuity and therefore (3)

|E(u) − E(Qu)| ≤ C ku − Quk2V .

From now on, the constant C may vary from line to line. By the definition of differentiability, (4)

E ′ (u) − E ′ (Qu) = L(Qu)(u − Qu) + o(ku − Quk).

We apply the projection I − P to this equality and use the definition of S in order to obtain (I − P )E ′ (u) = (I − P )L(Qu)(u − Qu) + o(ku − Quk). By the proof of Lemma 1, the operator (I − P )L(ϕ) : V1 → V1∗ is boundedly invertible. Hence, by continuity and if we choose U small enough, then (I − P )L(Qu) : V1 → V1∗ is boundedly invertible for all u ∈ U and the inverses are uniformly bounded in U. As a consequence, there exists a constant C ≥ 0 such that for every u ∈ U (5)

ku − QukV ≤ C k(I − P )E ′ (u)kV ∗ ≤ C kE ′ (u)kV ∗ .

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Combining the estimates (3) and (5) with the assumption that E|S satisfies the Lojasiewicz-Simon inequality in U ∩ S, we obtain that for every u ∈ U |E(u) − E(ϕ)| ≤ |E(u) − E(Qu)| + |E(Qu) − E(ϕ)| 1/(1−θ)

≤ C kE ′ (u)k2V ∗ + C kE ′ (Qu)kV ∗

.

By (4) and (5), kE ′ (Qu)kV ∗ ≤ kE ′ (u)kV ∗ + C ku − QukV ≤ C kE ′ (u)kV ∗ . As a consequence, for every u ∈ U, 1/(1−θ)

|E(u) − E(ϕ)| ≤ C kE ′ (u)k2V ∗ + kE ′ (u)kV ∗

.

Choosing U sufficiently small, we can by continuity assume that kE ′ (u)kV ∗ ≤ 1 for every u ∈ U. Since θ ∈ (0, 21 ], we then obtain 1/(1−θ)

|E(u) − E(ϕ)| ≤ C kE ′ (u)kV ∗

for every u ∈ U,

and this is the claim.

There are basically two important cases in which the assumption on E from Theorem 2 can be verified. The first one uses Lojasiewicz’ gradient inequality for real analytic functions on Rn , [28]. The second one is more elementary. Applications of both corollaries will be illustrated by our examples in the last section. Corollary 3. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold S as in Lemma 1. If S is analytic and if the restriction E|S is analytic, then E satisfies the Lojasiewicz-Simon inequality near ϕ. Proof. By assumption on the linearization L(ϕ) and by Lemma 1, S is finite dimensional. By Lojasiewicz’ classical result [28], the restriction E|S satisfies the Lojasiewicz-Simon inequality near ϕ. The claim follows from Theorem 2. Corollary 4. Let E ∈ C 2 (V ), let ϕ ∈ V be a stationary solution, and assume that E ′′ (ϕ) = L(ϕ) is a Fredholm operator. Define the critical manifold S as in Lemma 1. Assume that the set of stationary solutions S0 = {u ∈ V : E ′ (u) = 0} forms a neighbourhood of ϕ in S. Then E satisfies the Lojasiewicz-Simon inequality for the Lojasiewicz exponent θ = 21 .

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7

Proof. By assumption, the derivative E ′ is constant zero in a neighbourhood of ϕ in S. This implies that the restriction E|S is constant in the same neighbourhood. A constant function trivially satisfies the Lojasiewicz-Simon inequality for the Lojasiewicz exponent θ = 12 . The claim follows from Theorem 2.

3. Remarks Remark 5. The formulation and the proof of the Lojasiewicz-Simon inequality are independent of the Hilbert space H. In fact, the Hilbert space only allows us to identify the dual space V ∗ and thus also the derivative E ′ . Different choices of the Hilbert space H and different choices of the inner product in H lead to different identifications of the dual space V ∗ and the derivative E ′ ; see the examples in the next section. The value kE ′ (u)kV ∗ , however, does not change. The validity of the Lojasiewicz-Simon inequality is thus independent of the choice of H. Also the proof of the inequality can be formulated without using the Hilbert space H; simply identify V ∗ with V , or see [5]. In our proof the space H was only used for choosing a particular projection P . The Gelfand triple V ֒→ H = H ∗ ֒→ V ∗ is solely motivated by applications to partial differential equations. Remark 6. The critical manifold is in general not unique. The definition of the critical manifold S depends on the choice of the projection P . In order to simplify the presentation of the proof we have chosen the orthogonal projection (orthogonal in H) onto the kernel of the linearization L(ϕ). However, as remarked above, the space H and the inner product in H are not unique and thus also the projection P is not unique. Therefore, in general the critical manifold can vary when varying the projection P . On the other hand, the critical manifold always contains the set of stationary solutions. If the critical manifold coincides with the set of stationary solutions, then it is of course independent of the choice of the projection P . Remark 7. The critical manifold is not the center manifold. This follows already from the non-uniqueness of the critical manifold. However, both manifolds are tangent to the kernel of L(ϕ), and both manifolds have the same dimension than the kernel of L(ϕ). Both manifolds are critical in the following senses: in the case of existence of a center manifold, the difficult dynamics (different from exponential stability or exponential expansiveness) take place there. And by Theorem 2, the critical manifold is the set where the Lojasiewicz-Simon inequality may fail, but if the inequality is satisfied in the critical manifold then it is satisfied everywhere.

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Note, however, that if the critical manifold consists only of stationary solutions (at least locally) then the critical manifold and the center manifold do coincide and the dynamics are trivial in the center manifold. Remark 8. The Lojasiewicz-Simon inequality can be refined. Given E ∈ C 1 (V ) and ϕ ∈ V a stationary solution, we say that E satisfies the (generalized) Lojasiewicz-Simon inequality if there exists a neighbourhood U ⊂ V of ϕ and a strictly increasing function θ ∈ C([0, ∞[) ∩ C 1 (]0, ∞[) such that θ(0) = 0 and 1 ≤ kE ′ (u)kV ∗ . ′ θ (|E(u) − E(ϕ)|) This definition is due to Haraux & Jendoubi [20] and Huang [22]. The classical Lojasiewicz-Simon inequality is obtained by taking θ(t) = ctθ . Considering this type of inequality can be an advantage because of two reasons. First, there are more functionals which may satisfy the generalized Lojasiewicz-Simon inequality since we are allowing functions θ which are very slowly increasing. The proof of convergence to stationary solutions works also for this type of inequality. Second, it is known that the decay rate to stationary solutions depends on the function θ (or before: the Lojasiewicz exponent). Thus, also functions θ which are between two powers, like e.g. the function θ(t) = tα log(e + 1t ), are of interest. 4. Applications The applications which we describe in this section are all based on the same functional E! By this we want to show that the proof of the Lojasiewicz-Simon inequality is a problem which is independent of particular evolution equations. Let Ω ⊂ Rn be an open set and let f ∈ C 2 (Ω × R; R) be subcritical in the sense that ∂2f | 2 (x, s)| ≤ C|s|p for every s ∈ R, ∂s for some constant C ≥ 0 and some p ≥ 1 satisfying ( 4 if n ≥ 3, n−2 p< +∞ if n = 2. If n = 1, then we impose no growth condition on f . If Ω is bounded then the 2 growth condition on ∂∂sf2 for small s ∈ R can be relaxed to mere boundedness. Define the functional E : H01 (Ω) → R by Z Z 1 2 E(u) := |∇u| + F (x, u), 2 Ω Ω

THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES

where F (x, u) =

Ru 0

9

f (x, s) ds. It is an exercise to show that E is C 2 .

The functional E plays an important role in several applications. We mention some examples of gradient systems, i.e. systems which can abstractly be written in the form (1). Throughout these examples we put V = H01 (Ω). Example 9. Let H = L2 (Ω), equipped with the usual inner product. Then V embeds continuously and densely into H. When we identify H = L2 (Ω) with its dual, then V ∗ is the distribution space H −1(Ω) and E ′ (u) = −∆u + f (x, u). The evolution equation (1) thus becomes the semilinear heat equation ut − ∆u + f (x, u) = 0, subject to Dirichlet boundary conditions. The asymptotic behaviour of this semilinear heat equation has been studied in many articles. If Ω ⊂ R is a bounded interval, then every solution with relatively compact range in H01 converges to a stationary solution, [36], [29]. Zelenyak uses in his proof in [36] in principle the same idea than Lojasiewicz, while Matano’s proof in [29] is completely different. Zelenyak’s proof contains in addition a nice argument by contradiction the idea of which is as follows: if the solution does not converge then the ω-limit set is a connected continuum in the critical manifold. The critical manifold, however, is one dimensional by Lemma 1 and classical theory of ordinary differential equations. We thus find a point in the ω-limit set which is in the interior of the ω-limit set with respect to the critical manifold S. Near this point, the functional satisfies the LojasiewiczSimon inequality by Corollary 4. However, if the Lojasiewicz-Simon inequality holds near some point of the ω-limit set, then the solution must converge; a contradiction! This argument by contradiction has been extended by Haraux & Jendoubi in [19] and has been applied in many other situations including wave equations. If Ω ⊂ Rn is a bounded set and if f is analytic in the second variable, uniformly with respect to the first variable, then one may apply Corollary 3 in order to see that E satisfies the Lojasiewicz-Simon inequality near every stationary solution (and hence every bounded solution of the heat equation converges). Note, however, that in general E is not analytic on the energy space V = H01 (Ω) so that analyticity of the critical manifold S and of the restriction E|S do not follow from Lemma 1 directly. However, E is analytic on H01 (Ω) ∩ L∞ (Ω), and analyticity of S and of E|S is then deduced from elliptic regularity and arguments similar to those in the proof of Lemma 1; see e.g. [35], [25], [18].

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It is known that without any further conditions on f , solutions with relatively compact range in H01 (Ω) need not converge, [33], [34]. These counterexamples show that the Lojasiewicz-Simon inequality is in general not satisfied by the functional E. In the cases described above (i.e. when Ω is bounded) the Fredholm property of L(ϕ) follows from the Rellich-Kondrachov theorem and the spectral theory of compact operators. However, the Fredholm property can also be proved in special situations by using the spectral theory of Schr¨odinger operators. For example, if Ω = Rn , if f (x, s) = s − |s|p−1s for some p < n+2 , and if n−2 ϕ 6= 0 is a positive stationary solution (in fact, there exists only one up to translations), then L(ϕ) is Fredholm. Moreover, dim ker L(ϕ) = n and this is exactly the dimension of the set of positive stationary solutions which are obtained by translations of ϕ. Hence, the critical manifold (near a positive stationary solution) consists only of stationary solutions, and therefore the Lojasiewicz-Simon inequality holds by Corollary 4. These facts have been proved and used by Feireisl & Petzeltov´a [14] and Faˇsangov´a [11] in order to show that positive bounded solutions of the heat equation converge. See also [10] for a nonautonomous heat equation. Positivity is in general necessary for convergence as was shown by Faˇsangov´a & Feireisl [12]. Example 10. Let H = L2 (Ω), equipped with the inner product Z 1 (u, v)m := uv , m Ω

where m ∈ L∞ (Ω) is a positive function satisfying m1 ∈ L∞ (Ω). This inner product is clearly equivalent to the usual inner product. Still, the space V is dense in H and V ∗ = H −1 (Ω). However, due to the change of the inner product, the derivative E ′ becomes E ′ (u) = −m∆u + mf (x, u), and when we put g = mf , then the evolution equation (1) becomes the semilinear diffusion equation ut − m∆u + g(x, u) = 0. Example 11. Assume that Ω is bounded and let H = H01 (Ω), equipped R with the inner product (u, v)H01 = Ω ∇u∇v. This means in particular that we identify V with its dual V ∗ . With this identification, the derivative E ′ becomes E ′ (u) = u + (−∆)−1 f (x, u). The resulting gradient system is of particular interest in numerical computations. Here one is interested in solving the elliptic problem −∆u + f (x, u) = 0,

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11

i.e. in finding the stationary solutions of the functional E. Of course, these do not change with different identifications of V ∗ . The resulting evolution equation for the so-called Sobolev gradient of E above is a steepest descent method. Clearly, the gradient system obtained for this choice of H is in numerical computations approximated by a finite dimensional gradient system, but the advantage to approximate just this system and not for example the heat equation lies in the better regularity of the Sobolev gradient resulting in better convergence behaviour of the steepest descent method. For the theory of Sobolev gradients and the applications to numerics we refer to Neuberger [30], [31]. Example 12. Assume again that Ω is bounded, equip H01 (Ω) with the inner product from the preceeding example, and let H = H −1 (Ω), equipped with the inner product (u, v)H −1 = ((−∆)−1 u, (−∆)−1 v)H01 , where (−∆)−1 is the inverse of the negative Dirichlet-Laplace operator. In this example we have V ∗ = H −3 (Ω) and the derivative E ′ becomes E ′ (u) = ∆(∆u − f (x, u)), subject to the Dirichlet boundary conditions u|∂Ω = ∆u|∂Ω = 0. With this identification the evolution equation (1) results into the Cahn-Hillard equation ut + ∆(∆u − f (x, u)) = 0, subject to the above Dirichlet boundary conditions. It should be noted that these Dirichlet boundary conditions are not the usual ones for the CahnHilliard equation and thus this example is artificial. On the other hand, the more usual Neumann boundary conditions are obtained by simply considering the energy space V = H 1 (Ω) (or V the space of H 1 functions with zero mean) instead of H01 (Ω). Concerning the Lojasiewicz-Simon inequality everything then works similarly provided the set Ω is sufficiently regular so that H 1 (Ω) embeds compactly into L2 (Ω) (this ensures the Fredholm property of the linearization L(ϕ)). Convergence of bounded solutions of the Cahn-Hillard equation has been proved by Hoffmann & Rybka [21]. For the Cahn-Hillard equation with dynamic boundary conditions, see [7]. Example 13. We finally discuss an evolution equation which is not a gradient system. Take again H = L2 (Ω) as in Example 9 so that E ′ (u) = −∆u+f (x, u). The functional E plays an important role also for the semilinear damped wave equation utt + ut − ∆u + f (x, u) = 0, subject to Dirichlet boundary conditions. Jendoubi was the first to apply the Lojasiewicz-Simon inequality to prove convergence to stationary solutions in this gradient-like system, [24]; see also [18]. The convergence results are similar

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to those for the semilinear heat equation since Jendoubi’s proof of convergence mainly uses the Lojasiewicz-Simon inequality for the functional E. Therefore, if Ω ⊂ R is a bounded interval, then every weak solution which is bounded in H01 converges to a stationary solution. The same holds true if Ω ⊂ Rn is bounded and if f is analytic in the second variable, uniformly with respect to the first one. In general, bounded solutions need not converge, [26]. For a convergence result for wave equations on the whole space Rn , see [13]. References 1. S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-ﬁeld model with memory, J. Evolution Equations 1 (2001), 69–84. 2. S. Aizicovici and H. Petzeltov´a, Asymptotic behavior of solutions of a conserved phaseﬁeld system with memory, J. Integral Equations Appl. 15 (2003), 217–240. 3. H. Attouch, X. Goudou, and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: global exploration of the local minima of a real-valued function asymptotic by analysis of a dissipative dynamical system, Commun. Contemp. Math. 2 (2000), 1–34. 4. H. Cartan, Calcul diﬀ´erentiel, Hermann, Paris, 1967. 5. R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), 572–601. 6. R. Chill and E. Faˇsangov´a, Convergence to steady states of solutions of semilinear evolutionary integral equations, Calc. Var. Partial Diﬀerential Equations 22 (2005), 321–342. 7. R. Chill, E. Faˇsangov´a, and J. Pr¨ uss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr. (2006), to appear. 8. R. Chill and A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations, J. Diﬀerential Equations (2006), to appear. 9. R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Analysis, Ser. A: Theory Methods 53 (2003), 1017–1039. 10. , Convergence to steady states of solutions of non-autonomous heat equations in RN , J. Dynam. Diﬀerential Equations (2006), to appear. 11. E. Faˇsangov´a, Asymptotic analysis for a nonlinear parabolic equation on R, Comment Math. Univ. Carolin. 39 (1998), 525–544. 12. E. Faˇsangov´a and E. Feireisl, The long-time behaviour of solutions to parabolic problems on unbounded intervals: the inﬂuence of boundary conditions, Proc. Roy. Soc. Edinburgh 129A (1999), 319–329. 13. E. Feireisl, Long-time behavior and convergence for semilinear wave equations on RN , J. Dynam. Diﬀerential Equations 9 (1997), 133–155. 14. E. Feireisl and H. Petzeltov´a, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Diﬀerential Integral Equations 10 (1997), 181–196. 15. E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Diﬀerential Equations 12 (2000), 647–673. 16. A. Haraux, Syst`emes dynamiques dissipatifs et applications, Masson, Paris, 1990. 17. A. Haraux and M. A. Jendoubi, Convergence of solutions to second-order gradient-like systems with analytic nonlinearities, J. Diﬀerential Equations 144 (1998), 313–320.

THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES

18. 19. 20. 21. 22.

23. 24.

25. 26.

27.

28. 29. 30. 31. 32. 33. 34. 35. 36.

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Universit´ e Paul Verlaine - Metz, Laboratoire de Math´ ematiques et Apˆt. A, Ile du Saulcy, 57045 Metz plications de Metz et CNRS, UMR 7122, Ba Cedex 1, France E-mail address: [email protected]