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)
u̇(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
2
RALPH CHILL
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, Théorème 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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RALPH CHILL
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, Théorème 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, Théorème 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 ∗ .
6
RALPH CHILL
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 .
THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES
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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RALPH CHILL
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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RALPH CHILL
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 Schrödinger 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 & Petzeltová [14] and Fašangová [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 Fašangová & 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,
THE LOJASIEWICZ-SIMON GRADIENT INEQUALITY ON HILBERT SPACES
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
12
RALPH CHILL
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].
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Université Paul Verlaine - Metz, Laboratoire de Mathématiques et Applications de Metz et CNRS, UMR 7122, Bât. A, Ile du Saulcy, 57045 Metz
Cedex 1, France
E-mail address: [email protected]