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Long-time behavior of nonlinear integro-differential evolution equations
Nonlinear Analysis 91 (2013) 20–31
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Long-time behavior of nonlinear integro-differential evolution equations Justino Sánchez a , Vicente Vergara b,∗ a
Universidad de La Serena, Departamento de Matemáticas, Avenida Cisternas 1200, La Serena, Chile
b
Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta N. 1520, Arica, Chile
article
info
Article history: Received 21 January 2013 Accepted 12 June 2013 Communicated by S. Carl
abstract We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space. © 2013 Published by Elsevier Ltd
MSC: primary 37B25 37B35 secondary 45M05 35B40 Keywords: Integro-differential evolution equations Fractional derivative Gradient sytem Lyapunov function Convergence to steady state Lojasiewicz–Simon inequality
1. Introduction In this paper we study the long-time behavior of globally bounded strong solutions to nonlinear evolutionary equations with memory of the form d dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds + k∞ u(t ) + E ′ (u(t )) = f (t ), 0
t > 0,
(1.1)
u(0) = u0 , in a real Hilbert space H. Here E ′ is the Fréchet derivative of a functional E ∈ C 1 (V ), where V is a Hilbert space which densely and continuously injects into H. The vector-valued function f and the scalar kernel k1 are locally integrable functions on R+ ; k0 , and k∞ are real constants, and k0 is chosen nonnegative. The vector u0 ∈ V stands for the initial condition for the unknown function u. Under suitable conditions on k1 and f , we prove that any globally bounded solution of (1.1) converges to a steady state in V , that is, u∞ := limt →∞ u(t ) exists in V and u∞ is a solution of the problem k∞ u∞ + E ′ (u∞ ) = 0,
∗
Corresponding author. E-mail addresses: [email protected] (J. Sánchez), [email protected] (V. Vergara).
0362-546X/$ – see front matter © 2013 Published by Elsevier Ltd http://dx.doi.org/10.1016/j.na.2013.06.006
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
21
provided that the solution u has relatively compact range in V and the functional E satisfies the Łojasiewicz–Simon inequality (see Definition 3). Note that, without loss of generality, we may set k∞ = 0 in (1.1) since we can include it into the nonlinear term E by defining a new nonlinearity k∞
E˜ (u) =
2
|u|2H + E (u).
In the literature one finds many papers for problems of the form (1.1), as well as variants of them which are studied in a strong setting, assuming smoothness on the nonlinearities. See e.g. [1–5]. It seems that results about convergence to steady state of globally bounded solutions of (1.1) under the general setting given above are unknown, except for some special cases, where convergence results are well-known. Examples of the latter are the following. In case k0 = 1 and k1 = k∞ = 0, Eq. (1.1) becomes a first order problem of the form ut + E ′ ( u) = f ,
t > 0,
u(0) = u0 , which has been studied by many authors; see e.g. [6–8]. See also [9]. The case k0 = k∞ = 0 and k1 ̸= 0 of Eq. (1.1) becomes a problem of the form d
t
dt
k1 (t − s)(u − u0 )(s)ds
+ E ′ (u(t )) = f (t ),
t > 0,
(1.2)
0
u(0) = u0 , which has been studied by Vergara and Zacher [10] in a finite-dimensional Hilbert space. In this paper we extend the results obtained in [10] to real Hilbert spaces for Eq. (1.1). More precisely, we study integrodifferential evolution equations of two types. – Problems of order 1 (k0 > 0) with memory (k1 ̸= 0): d
dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds
+ E ′ (u(t )) = f (t ),
t > 0,
0
u(0) = u0 . – Problems of order less than 1 (k0 = 0): d
t
dt
k1 (t − s)(u(s) − u0 )ds
+ E ′ (u(t )) = f (t ),
t > 0,
0
u(0) = u0 . 1 ,2
Concerning applications, let Ω be a bounded domain in Rn with smooth boundary ∂ Ω . Setting H = L2 (Ω ), V = W0 (Ω ) (the classical Sobolev space) and
E (v) =
1
2
|∇v|2 dx +
Ω
Ω
v ∈ W01,2 (Ω ),
G(x, v)dx,
problem (1.1) becomes a problem of the form d dt
k0 u(t ) +
u = 0,
t
k1 (t − s)(u(s) − u0 )ds
+ k∞ u(t ) − ∆u(t ) + g (x, u(t )) = f (t ),
in R+ × Ω ,
0
on R+ × ∂ Ω ,
u(0) = u0 ,
in Ω ,
(1.3)
where g (x, s) := ∂s G(x, s) for all s ∈ R, which arises as a model for nonlinear heat flow in material with memory; see e.g., Gurtin and Pipkin [11], MacCamy [12], Nunziato [13], and the monograph Prüss [4]. See also [2], where existence and uniqueness results of globally bounded solutions of (1.3) were obtained for more general functions g. Concerning convergence to steady state for nonlinear equations with memory, there has been partial progress; see e.g. [14–20]. The reason for this lies essentially in the fact that these problems do not generate in general a semiflow in the natural phase space. Another technical difficulty consists in proving that the solutions of such problems are globally bounded and have relatively compact range in a natural phase space. Further, for problems with memory, it is in general a highly nontrivial task to construct Lyapunov functions (cf. [21, Chapter 14] and [17]) which are appropriate to investigating the asymptotic behavior of globally bounded solutions. The paper [10] develops a method for finding Lyapunov functions for problems of order less than 1, and for orders between 1 and 2 (in time), in finite-dimensional Hilbert spaces, which combined with the Łojasiewicz inequality allows one to prove convergence to a single steady state of bounded solutions of such problems.
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Recently, the second author [22] extended these results to real Hilbert spaces for problems of order between 1 and 2. The purpose of this paper is to deal with problems of order less than 1 and of order 1 with memory, in order to close this gap. In this article we show that any globally bounded strong solution of (1.1) (see Definition 2) converges to a single steady state. Here we make use of ideas and results obtained by Vergara and Zacher [10] and Vergara [22]. By means of a suitable approximation technique our convergence result can properly be formulated to any globally bounded weak solution. This will be treated in a separate work. The proof of our main results Theorems 3.1 and 4.1 concerning the convergence to the steady state is based on an appropriate new Lyapunov function, compactness properties, and on the Łojasiewicz–Simon inequality. The paper is organized as follows. Section 2 recalls some basic definitions and collects important properties of the operator Bv = dtd (k ∗ v) with nonnegative, nonincreasing kernel k, including the key result from [10]. In Section 3 we study problems of order 1 with memory and we state the main result Theorem 3.1 in this section. In Section 4 we discuss problems of order less than 1, in which we state the corresponding result Theorem 4.1. 2. Preliminaries Let V and H be real Hilbert spaces such that V is densely and continuously embedded into H. We shall identify H with its dual H ′ , that is, we have V ↩→ H ≈ H ′ ↩→ V ′ . We denote by ⟨·, ·⟩ the scalar products and duality relations; these will be made precise by subscripts according to the respective space. The operator E ′ is nonlinear and continuous from V into V ′ , and it is the Fréchet derivative of a functional E ∈ C 1 (V ). We will use the standard notation for the time derivative of a function u, that is, dtd u = u˙ = ut . For an interval J := [0, T ] (T > 0), s > 0 and 1 < p < ∞, we denote by Hps (J ; H ) the vector-valued Bessel potential space. We recall that if s ∈ N then Hps (J ; H ) stands for the classical Sobolev space. For s ∈ (0, 1], we denote by 0 Hps (J ; H ) the subspace of all functions f in Hps (J ; H ) with f (0) = 0 provided this trace exists; see e.g. [23]. The convolution of two functions a, b ∈ L1,loc (R+ ) (R+ := [0, ∞)) is defined by (a ∗ b)(t ) = t ≥ 0.
t 0
a(t − s)b(s)ds,
Lemma 2.1. Let H be a real Hilbert space and T > 0. Then for each k ∈ H11 ([0, T ]) and each v ∈ L2 ([0, T ]; H ) we have
d dt
(k ∗ v)(t ), v(t )
= H
1 d 2 dt
+
(k ∗ |v|2H )(t ) +
2
k(t )|v(t )|2H
t
1
1 2
[−k˙ (s)] |v(t ) − v(t − s)|2H ds,
for a.a. t ∈ (0, T ).
(2.1)
0
See [10, Lemma 2.2] for the proof. Next, we state a recent result due to Vergara and Zacher [10] which is the key to finding a proper Lyapunov function for Eq. (1.1). Theorem 2.2. Let H be a real Hilbert space, T > 0, and a ∈ L1,loc (R+ ) be nonnegative and nonincreasing such that k ∗ a = 1 in (0, ∞) for some nonnegative k ∈ L1,loc (R+ ). Suppose that v ∈ L2 ([0, T ]; H ) and that there exists x ∈ H such that a ∗ (v − x) ∈ 0 H21 ([0, T ]; H ), as well as a ∗ |v − x|2H ∈ 0 H11 ([0, T ]). Then
d dt
1 d 1 (a ∗ v)(t ), v(t ) ≥ (a ∗ |v|2H )(t ) + a(t )|v(t )|2H , H
2 dt
2
for a.a. t ∈ (0, T ).
(2.2)
The assumption a ∗ |v − x|2H ∈ 0 H11 ([0, T ]) in Theorem 2.2 seems a little awkward from the point of view of applications. We will now describe a rather wide class of kernels a for which we can guarantee that this assumption holds. ˆ In what follows ∞f stands for the Laplace transform of f . A function a ∈ L1,loc (R+ ) is said to be of subexponential growth if, for all ε > 0, 0 e−εt |a(t )| dt < ∞. Following [4, Definition 3.3] we say that a kernel a ∈ L1,loc (R+ ) of subexponential growth is 1-regular if there is a constant c > 0 such that |λˆa′ (λ)| ≤ c |ˆa(λ)| for all Reλ > 0. A kernel a ∈ L1,loc (R+ ) of subexponential growth satisfying aˆ (λ) ̸= 0, Re λ > 0, is called θ -sectorial (θ > 0) if | arg aˆ (λ)| ≤ θ for all Re λ > 0 (cp. [4, Definition 3.2]). The following class of kernels has been introduced in [24, Definition 2.6.3]. See also [25, Definition 2.1]. Definition 1. Let a ∈ L1,loc (R+ ) be of subexponential growth, and let θa > 0 and α ≥ 0. Then a is said to belong to the class K 1 (α, θa ) if a is 1-regular and θa -sectorial, and satisfies lim sup | aˆ (µ)| µα < ∞, µ→∞
lim inf | aˆ (µ)| µα > 0, µ→∞
We have now the following result.
and
lim inf | aˆ (µ)| > 0. µ→0
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
23
Proposition 2.3. Let H be a real Hilbert space, T > 0, and a ∈ L1,loc (R+ ) be such that k ∗ a = 1 in (0, ∞) for some k ∈ K 1 (α, θ ) with α ∈ (0, 1) and θ < π . Suppose that v ∈ L2 ([0, T ]; H ) and a ∗ v ∈ 0 H21 ([0, T ]; H ). Then we have a ∗ |v|2H ∈ 0 H11 ([0, T ]). For the proof of this proposition see [10, Proposition 2.1] and [25, Corollary 2.1]. Definition 2. A function u ∈ C (J ; V ) is called (a) a strong solution of (1.1) if u ∈ H21 (J ; H ) ∩ C (J ; V ) and (1.1) holds a.e. on J; (b) a globally bounded strong solution of (1.1) if u is a strong solution on each interval J and u ∈ L∞ (R+ ; V ). Henceforth we simply write ‘‘solution’’ instead of ‘‘strong solution’’. Definition 3. We say that a functional E ∈ C 1 (V ) satisfies the Łojasiewicz–Simon inequality in the neighborhood of some point ϑ ∈ V if there exist constants θ ∈ (0, 1/2], C > 0, and σ > 0 such that, for all u ∈ V with |u − ϑ|V ≤ σ , we have
|E (u) − E (ϑ)|1−θ ≤ C |E ′ (u)|V ′ . Here the constant θ is called the Łojasiewicz exponent. A typical example of a functional E is given by
E (u) =
1 2
η(u, u) +
Ω
ϕ(x, u)dx,
u ∈ V,
where η : V × V → R is a bilinear, continuous, symmetric and coercive form, and the nonlinear term ϕ(x, ·) belongs to C 2 (V ) for a.a. x ∈ Ω . We refer the reader to [6,8] for a comprehensive study of this subject. For every globally bounded solution u of (1.1) we define the ω-limit set by
ω(u) = {ϑ ∈ V : there exists (tn ) ↗ ∞ s.t. u(tn ) → ϑ in V } . 3. Problems of order 1 with memory In this section we investigate problems of the form d dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds + E ′ (u(t )) = f (t ),
t > 0,
u(0) = u0 ,
(3.1)
0
where k0 is a positive constant and the following assumptions are satisfied. Throughout the paper we set v := u0 )).
d dt
(k1 ∗ (u −
(A1) k1 ∈ L1,loc (R+ ) is nonnegative and nonincreasing. (A2) There is a nonnegative nonincreasing kernel a ∈ L1,loc (R+ ) such that
(k1 ∗ a)(t ) = 1,
t > 0.
(A3) There exists a constant γ > 0 such that the solution ζ of
ζ (t ) + γ
t
ζ (s)ds = a(t ),
t > 0,
(3.2)
0
is nonnegative. (A4) f ∈ L1 (R+ ; H ) ∩ L2 (R+ ; H ) is such that the function h(t ) :=
∞
∞
t
|f (τ )|2H dτ
21 ds
s
is bounded on R+ .
(AE ) the functional E : V → R belongs to C 1 (V ). Remark 1. (a) Typical examples of kernels k1 and a, which satisfy (A1)–(A3) are given by k1 (t ) = g1−α (t )e−γ t ,
(3.3)
and a(t ) = gα (t )e−ωt + γ (1 ∗ [gα e−γ · ])(t ),
t > 0, α ∈ (0, 1), γ > 0,
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
respectively, where gβ (t ) = t β−1 /Γ (β), t > 0, β > 0, and Γ (·) is the Gamma function. Furthermore, we have that a ∈ K 1 (α, (1 + α)π /2) and k1 ∈ K 1 (α, απ /2); see [24, Example 2.1]. (b) For each γ > 0 the unique solution of (3.2) is given by
ζγ (t ) := a(t ) − γ (e−γ · ∗ a)(t ),
t > 0.
Hence, if condition (A3) holds, by decreasing γ , we may assume that ζγ is strictly positive and strictly decreasing on (0, ∞). ∞ Furthermore, ζ ∈ L1 (R+ ) and limt →∞ ζ (t ) = 0, therefore a∞ := limt →∞ a(t ) = γ 0 ζ (s)ds > 0; see [10, Remark 3.1]. (c) Assumptions (A1)–(A3) imply that k1 ∈ L1 (R+ ) with |k1 |L1 ≤ 1/a∞ . (d) Assumption (Af) was first introduced by Huang and Takáč in [9, Remark 3.1] to prove convergence to equilibrium of bounded solutions for a nonlinear heat equation with a time-dependent perturbation f . (e) Under the condition (Af) the function h belongs to C01 (R+ ). In particular, if f has polynomial or exponential decay to zero as time tends to infinity then h has the same property. (f) Let u be a solution of (3.1). Then a ∗v = u − u0 ∈ 0 H21 (J ; H ) by assumption (A2) and v ∈ L2 (J ; H ) by (Af)–(AE ) and (3.1). Hence if we assume that a ∈ K 1 (α, θa ), we obtain by Proposition 2.3 that a ∗ |v|2H ∈ 0 H11 (J ). Thus we can apply inequality (2.2) in Theorem 2.2 to the function v . Now we establish our first main result on convergence of globally bounded solutions of (3.1). Theorem 3.1. Let u be a globally bounded solution of (3.1) with k0 > 0, u0 ∈ V , and assume that a ∗ |v|2H ∈ 0 H11,loc (R+ ). Suppose that (i) the set {u(t ) : t ≥ 0} is relatively compact in V ; (ii) (A1)–(A3), (Af), and (AE ) hold; (iii) E satisfies the Łojasiewicz–Simon inequality near each point ϑ ∈ ω(u) ⊂ V . Then limt →∞ u(t ) = ϑ in V , and ϑ is a stationary solution, i.e. E ′ (ϑ) = 0. Before beginning the proof of Theorem 3.1, we state some auxiliary facts about properties of the ω-limit set ω(u) as well as of u itself. The proof of the following result shows how one may construct a proper Lyapunov function for Eq. (3.1). Proposition 3.2. Let u be a globally bounded solution of Eq. (3.1) and let ϵ ∈ (0, 2k0 ). Assume that (A1)–(A3), (Af), and a ∗ |v|2H ∈ 0 H11,loc (R+ ) hold. Then the function V0 : R+ → R defined by V0 ( t ) =
1 2
(ζ ∗ |v| )(t ) + E (u(t )) + 2 H
1
2ϵ
∞
|f (s)|2H ds,
t > 0,
(3.4)
t
is locally absolutely continuous and nonincreasing on R+ . Furthermore,
−
d dt
V0 (t ) ≥ k0 −
ϵ 2
|ut |2H +
γ 2
ζ ∗ |v|2H +
1 2
a∞ |v(t )|2H ≥ 0,
for a.a. t > 0.
(3.5)
Proof. We first multiply (3.1) by ut and obtain k0 |ut |2H + ⟨v, ut ⟩H +
d dt
E (u) = ⟨f , ut ⟩H .
(3.6)
Next we estimate separately the terms ⟨v, ut ⟩H and ⟨f , ut ⟩H . Note that from (A2) we get
⟨v, ut ⟩H =
d dt
k1 ∗ (u − u0 ), ut
= H
d dt
a ∗ v, v
,
(3.7)
H
and by Theorem 2.2 and (A3) we have
d dt
a ∗ v, v
≥ H
≥
1 d 2 dt 1 d 2 dt
a ∗ |v|2H +
ζ ∗ |v|2H +
1 2
γ 2
a(t )|v(t )|2H
ζ ∗ |v|2H +
1 2
a∞ |v(t )|2H .
(3.8)
Next we simply estimate the term ⟨f , ut ⟩H by Young’s inequality, i.e.
⟨f , ut ⟩H ≤
1 2ϵ
ϵ |f |2H + |ut |2H .
(3.9)
2
Replacing (3.7)–(3.9) by (3.6) we obtain (3.5) with V0 defined by (3.4).
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
25
Proposition 3.3. Let u be a globally bounded solution of Eq. (3.1). Assume that (A1)–(A3), (Af)–(AE ) hold, a ∗ |v|2H ∈ 1 0 H1,loc (R+ ), and the set {u(t ) : t ≥ 0} is relatively compact in V . Then: (i) ut , v ∈ L2 (R+ ; H ) and ζ ∗ |v|2H ∈ L1 (R+ ) ∩ C0 (R+ ); (ii) the potential E is constant on ω(u) and limt →∞ E (u(t )) exists; (iii) for every ϑ ∈ ω(u) we have E ′ (ϑ) = 0. Proof. Since V0 defined by (3.4) is absolutely continuous and decreasing on R+ and E (u) is bounded from below, we have that V0∞ := limt →∞ V0 (t ) exists. Thus integrating (3.5) we prove (i) except that ζ ∗ |v|2H ∈ C0 (R+ ). This will be shown later. Since u has relatively compact range in V , it follows that the ω(u) is nonempty, compact and connected. Let ϑ ∈ ω(u) and choose tn ↗ ∞ such that u(tn ) → ϑ in V . Since ut ∈ L2 (R+ ; H ), we obtain u(tn + s) = u(tn ) +
tn + s
ut (τ )dτ → ϑ
in H , for every s ∈ [0, 1].
tn
This, together with the relative compactness of the trajectory, implies that u(tn + s) → ϑ in V for every s ∈ [0, 1]. Therefore, limn→∞ E (u(tn + s)) = E (ϑ) for every s ∈ [0, 1], and thus, by the dominated convergence theorem, 1
E (ϑ) = lim
n→∞
E (u(tn + s))ds. 0
In addition, integrating V0 (tn + ·) defined in (3.4) over [0, 1], we obtain
E (ϑ) + lim
n→∞
tn + 1
1 2
tn
(ζ ∗ |v| )(s) + 2 H
1 2ϵ
∞
|f (τ )|
2 Hd
n→∞
s
t
+1
1
τ ds = lim ∞
V0 (tn + s)ds = V0∞ . 0
1 Now, from assumption (Af) we obtain that limn→∞ t n |f (τ )|2H dτ ds = 0, and since ζ ∗ |v|2H ∈ L1 (R+ ) we get 2ϵ s n E (ϑ) = V0∞ , that is, E is constant on ω(u). Further, as a consequence of the above, we obtain that (ζ ∗ |v|2H )(t ) → 0 as t → ∞. Indeed, if the opposite were true then there would exist a δ > 0 and a sequence tn → ∞ as n → ∞ such that (ζ ∗ |v|2H )(tn ) ≥ δ for all n ∈ N. By compactness, there exists a subsequence tnk such that E (u(tnk )) → V0∞ as k → ∞, hence (ζ ∗ |v|2H )(tnk ) → 0 as k → ∞, which is a contradiction. Hence (ζ ∗ |v|2H )(t ) → 0 as t → ∞. Moreover, we see that limt →∞ E (u(t )) = V0∞ . Hence the claim (ii) is proved and the proof of (i) is complete. Next, since E ∈ C 1 (V ), we have E ′ (u(tn +s)) → E ′ (ϑ) in V ′ for every s ∈ [0, 1]. Further, using the dominated convergence theorem, Eq. (3.1), (i), and assumption (Af) we obtain
E ′ (ϑ) = lim
n→∞
tn + 1
E ′ (u(s))ds
tn
= lim − n→∞
tn + 1
(k0 ut (s) + v(s)) ds +
tn
tn + 1
f (s)ds = 0.
tn
The proof of the proposition is now complete.
3.1. Proof of Theorem 3.1 Let ϑ ∈ ω(u) and define a new energy function V1 : R+ → R by V1 (t ) := V0 (t ) − E (ϑ),
t > 0.
(3.10)
Thus V1 is nonnegative, nonincreasing, locally absolutely continuous on R+ , and limt →∞ V1 (t ) = 0. In addition, we may assume that V1 (t ) > 0 for all t > 0. This is because, if there exists a t > 0 such that V1 (t ) = 0, then V1 (s) = 0 for all s ≥ t and in this case u(t ) = ϑ for all t > 0. Now we will use our main assumption (iii). Let ϑ ∈ ω(u), since E is constant on ω(u), it follows from assumption (iii) and compactness of the ω-limit set that there exists an open set U ⊂ V such that ω(u) ⊂ U, and there are constants θ ∈ (0, 1/2] and C > 0 such that the inequality
|E (u(t )) − E (ϑ)|1−θ ≤ C |E ′ (u(t ))|V ′
(3.11)
holds for every u(t ) ∈ U. Further, since limt →∞ dist(u(t ), ω(u)) = 0, we see that there exists t ∗ ≥ 0 such that u(t ) ∈ U for all t ≥ t ∗ and (3.11) holds. Next, we compute and estimate the time derivative of V1 (t )θ . By (3.10) we obtain
V1 (t )
1−θ
≤ C1 |E (u(t )) − E (ϑ)|
1−θ
+ (ζ ∗ |v| )
2 1−θ H
∞
|f (s)|
1−θ
2 H ds
+ t
2 21 2(1−θ) H
≤ C2 |E (u(t ))|V ′ + (ζ ∗ |v| ) ′
∞
|f (s)|
21 2(1−θ )
2 H ds
+ t
.
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Since, 2(1 − θ ) ≥ 1 and (1 − θ )/θ ≥ 1 for all θ ∈ (0, 1/2], we conclude that
V1 ( t )
1−θ
2 21 H
≤ C3 |E (u(t ))|V ′ + (ζ ∗ |v| ) + ′
∞
|f (s)|
2 H ds
21
.
(3.12)
t
Therefore, from (3.5) and (3.12), we have
−
d dt
d V0 (t ) dt θ C0 |ut |2H + ζ ∗ |v|2H + |v|2H
[V1 (t )θ ] = −θ V1 (t )θ−1 ≥
12 2 | f ( s )| ds H t 2 θ C0 |ut |H + ζ ∗ |v|2H + |v|2H ≥ ∞ 1 1 C˜ 4 |ut |H + (ζ ∗ |v|2H ) 2 + |v|H + |f (t )|H + t |f (s)|2H ds 2 1
C4 |E ′ (u(t ))|V ′ + (ζ ∗ |v|2H ) 2 +
2 21 H
≥ C5 |ut |H + (ζ ∗ |v| ) + |v|H
∞
˜ − C5 |f (t )|H +
∞
|f (s)|
21
2 H ds
.
(3.13)
t
This in turn implies that ut ∈ L1 ([t ∗ , ∞), H ). Therefore limt →∞ u(t ) exists in H. Hence, from the relative compactness of u(t ) in V , our claim follows. 4. Problems of order less than 1 In this section we investigate problems of the form d dt
t
k1 (t − s)(u(s) − u0 )ds + E ′ (u(t )) = f (t ),
t > 0,
u(0) = u0 .
(4.1)
0
We will assume that the following assumptions hold. (A1) k1 ∈ L1,loc (R+ ) is nonnegative and nonincreasing. (A2) There is a nonnegative nonincreasing kernel a ∈ L1,loc (R+ ) such that
(k1 ∗ a)(t ) = 1,
t > 0.
(A3) There exists a constant γ > 0 such that the solution ζ of
ζ (t ) + γ
t
ζ (s)ds = a(t ),
t > 0,
0
is nonnegative.
(Af′ ) f ∈ H11 (R+ ; H ) ∩ H21 (R+ ; H ) is such that the function 21 ∞ ∞ ds h˜ (t ) := |f (τ )|2H + |f˙ (τ )|2H dτ t
s
is bounded on R+ . (AE ) the functional E : V → R belongs to C 1 (V ). Remark 2. (a) Assumptions (A1)–(A3) and (AE ) are the same as above, while (Af′ ) is clearly different from (Af). This new condition (Af′ ) was first used by Vergara and Zacher [10] to get convergence results for (4.1) in finite dimensional spaces. (b) Under the condition (Af′ ) the function h˜ belongs to C01 (R+ ). In particular, if f and f˙ have polynomial decay or exponential decay to zero as time tends to infinity then h˜ has the same property. t (c) Let u be a solution of (4.1) and set v(t ) := 0 k1 (t − s)(u(s) − u0 )ds as in the above section. Then a ∗ v = u − u0 ∈ 1 ′ 1 0 H2 (J ; H ) by assumption (A2) and v belongs to L2 (J ; H ) by (Af )–(AE ) and (4.1). Hence if we assume that a ∈ K (α, θa ), we 2 1 obtain by Proposition 2.3 that a ∗ |v|H ∈ 0 H1 (J ). Thus we can apply inequality (2.2) in Theorem 2.2 to the function v . (d) Notes (a)–(c) in Remark 1 still hold. Now we establish our main result on convergence of globally bounded solutions of (4.1). Theorem 4.1. Let u be a globally bounded solution of (4.1), u0 ∈ V , and assume that a ∗ |v|2H ∈ 0 H11,loc (R+ ). Suppose that (i) the set {u(t ) : t ≥ 0} is relatively compact in V ;
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
27
(ii) (A1)–(A3), (Af′ ), and (AE ) hold; (iii) E satisfies the Łojasiewicz–Simon inequality near each point ϑ ∈ ω(u) ⊂ V . Then limt →∞ u(t ) = ϑ in V , and ϑ is a stationary solution, i.e. E ′ (ϑ) = 0. Before beginning the proof of Theorem 4.1, we state some auxiliary facts about properties of the ω-limit set ω(u) as well as of u itself. It uses the following elementary lemma which will be used in the sequel (for the proof of this lemma, see for instance [10]). Lemma 4.2. Let H be a real Hilbert space and T > 0. Suppose that k ∈ L1,loc (R+ ) is nonnegative. Then for any v ∈ L2 ([0, T ]; H ) we have
|(k ∗ v)(t )|2H ≤ (k ∗ |v|2H )(t ) (1 ∗ k)(t ),
for a.a. t ∈ (0, T ).
The proof of the following result shows how one may construct a Lyapunov function for Eq. (4.1). Proposition 4.3. Let u be a globally bounded solution of Eq. (3.1). Assume that (A1)–(A3), (Af′ ) –(AE ) hold, and that a ∗ |v|2H ∈ 1 −1 }. Then the function V0 : R+ → R defined by 0 H1,loc (R+ ). Let M = 2|ζ |L1 (R+ ) max{γ , γ V0 (t ) =
1 2
(ζ ∗ |v|2H )(t ) + E (u(t )) − ⟨f (t ), (ζ ∗ v)(t )⟩H + M
∞
|f (τ )|2H + |f˙ (τ )|2H dτ ,
t ≥ 0,
(4.2)
t
is absolutely continuous and decreasing on R+ . Furthermore,
−
d dt
1
V0 (t ) ≥
2
a∞ |v(t )|2H +
γ 4
(ζ ∗ |v|2H )(t ) ≥ 0,
for a.a. t > 0.
(4.3)
Proof. We take the inner product of the integro-differential equation in (4.1) with ut to find that
⟨v, ut ⟩H +
d dt
E (u) = ⟨f , ut ⟩H ,
t > 0.
(4.4)
Hence, by (A2) we may write d
ut =
dt
( u − u0 ) =
d2 dt 2
[k1 ∗ a ∗ (u − u0 )] =
d dt
(a ∗ v),
(4.5)
which together with (4.4) yields
v,
d dt
d (a ∗ v) + E (u) = ⟨f , ut ⟩H , H
dt
t > 0.
(4.6)
Since v = −E ′ (u) + f ∈ L2,loc (R+ , H ) by the assumptions (AE ) and (Af′ ), a ∗ v = u − u0 ∈ 0 H21,loc (R+ ; H ) by the definition of a solution. Hence, we may apply inequality (2.2) to the first term in (4.6) to get the result d 1 dt
2
1 (a ∗ |v|2H )(t ) + E (u(t )) ≤ − a(t )|v(t )|2H + ⟨f (t ), ut (t )⟩H , 2
t > 0.
(4.7)
Using assumption (A3) we obtain from (4.7) that d dt
1 2
1 γ (ζ ∗ |v|2H )(t ) + E (u(t )) ≤ − a(t )|v(t )|2H − (ζ ∗ |v|2H )(t ) + ⟨f (t ), ut (t )⟩H , 2
2
t > 0.
(4.8)
In order to treat the term ⟨f , ut ⟩H , note that by Lemma 4.2 and Young’s inequality we have
⟨f , ut ⟩H = f , = ≤ ≤
d dt d dt d dt
d dt
(a ∗ v)
= f, H
d dt
(ζ ∗ v) + γ ⟨f , ζ ∗ v⟩H H
⟨f , ζ ∗ v⟩H − ⟨f˙ , ζ ∗ v⟩H + γ ⟨f , ζ ∗ v⟩H ⟨f , ζ ∗ v⟩H + 2|ζ |L1 (R+ ) γ |f |2H + γ −1 |f˙ |2H + γ ⟨f , ζ ∗ v⟩H + M |f |2H + |f˙ |2H + ζ ∗ |v|2H , 4
γ 4|ζ |L1 (R+ ) t > 0.
|ζ ∗ v|2H (4.9)
28
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Hence, we obtain from (4.8) and (4.9) that the estimate
−
d dt
V0 (t ) ≥
1 2
holds in the a.e. sense.
a∞ |v(t )|2H +
γ 4
(ζ ∗ |v|2H )(t ),
t >0
(4.10)
We consider now globally bounded solutions of (4.1) with relative compact orbit in V . For any such solution u, the ω-limit set ω(u) is nonempty, compact and connected. Proposition 4.4. Let u be a globally bounded solution of (4.1). Suppose that (A1)–(A3), (AE )–(Af′ ) hold, and that a ∗ |v|2H ∈ 1 0 H1,loc (R+ ). Then: (i) v ∈ L2 (R+ ; H ) and ζ ∗ |v|2H ∈ L1 (R+ ) ∩ C0 (R+ ); (ii) the potential E is constant on ω(u) and limt →∞ E (u(t )) exists; (iii) for every ϑ ∈ ω(u) we have E ′ (ϑ) = 0. Proof. Let u be a globally bounded solution of (4.1). Clearly, E (u) is bounded from below. Furthermore, by Young’s inequality and Lemma 4.2,
⟨f (t ), (ζ ∗ v)(t )⟩H ≤ |ζ |L1 (R+ ) |f (t )|2H +
1 4
(ζ ∗ |v|2H )(t ),
t ≥ 0,
and so it is clear that V0 : R+ → R defined in (4.2) is bounded from below. By Proposition 4.3, the function V0 is nonincreasing and therefore limt →∞ V0 (t ) = inft ≥0 V0 (t ) =: V∞ exists. The first part of assertion (i) follows then directly from estimate (4.10). In order to see that ζ ∗ |v|2H ∈ C0 (R+ ), note first that |v|L∞ (R+ ;H ) ≤ C for some constant C > 0. In fact, this follows from the equality v = −E ′ (u) + f and the boundedness of u and f . Now, given ε > 0 we can choose δ > 0 so that |ζ |L1 (0,δ) C 2 ≤ ε . Define the function ζδ by ζδ (t ) = ζ (t ), t ∈ (0, δ), and ζδ (t ) = 0, t ≥ δ . Using Young’s inequality we may then estimate
(ζ ∗ |v|2H )(t ) ≤ |ζδ ∗ |v |2H |L∞ (R+ ) + ((ζ − ζδ ) ∗ |v|2H )(t ) ≤ ε + ((ζ − ζδ ) ∗ |v|2H )(t ),
t ≥ 0.
(4.11)
Since 0 ≤ (ζ − ζδ )(t ) ≤ ζ (δ) for a.a. t > 0, we further have
|((ζ − ζδ ) ∗ |v |2H )(t )| =
t /2
(ζ − ζδ )(s)|v(t − s)|2H ds +
0
≤ ζ (δ)
t
t /2
|v(s)|2H ds + C 2
t t /2
(ζ − ζδ )(s)|v(t − s)|2H ds
t
t /2
(ζ − ζδ )(s) ds,
which together with |v(·)|2H ∈ L1 (R+ ) and ζ ∈ L1 (R+ ) shows that limt →∞ ((ζ − ζδ ) ∗ |v|2H )(t ) = 0. The latter property, together with (4.11) and the fact that ε can be chosen arbitrarily small, imply that limt →∞ (ζ ∗ |v|2H )(t ) = 0. Hence (i) is established. Now let ϑ ∈ ω(u) and tn ↗ ∞ be such that limn→∞ u(tn ) = ϑ in V . Since u(t ) − u0 = (a ∗ v)(t ),
t ≥ 0,
(4.12)
cf. (4.5), it follows that limm→∞ (a ∗ v)(tm ) exists and that
ϑ = u0 + lim (a ∗ v)(tm ).
(4.13)
m→∞
Let again a∞ := limt →∞ a(t ) > 0 and a˜ := a − a∞ . Using (4.12) and (4.13) we have, for tn ≤ t with n ∈ N, u(t ) − ϑ = (a ∗ v)(t ) − lim (a ∗ v)(tm ) m→∞
= (a ∗ v)(t ) − (a ∗ v)(tn ) + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) m→∞ t = (˜a ∗ v)(t ) − (˜a ∗ v)(tn ) + a∞ v(τ ) dτ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) . tn
Observe that 0 ≤ a˜ (t ) = ζ (t ) − γ
∞
ζ (τ ) dτ ≤ ζ (t ), t
t > 0,
m→∞
(4.14)
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
29
and thus, by Lemma 4.2, 1/2
|(˜a ∗ v)(t )|H ≤ [(ζ ∗ |v|2H )(t )]1/2 |ζ |L1 (R+ ) .
(4.15)
From (4.15) and (i) we deduce that lim (˜a ∗ v)(t ) = 0.
(4.16)
t →∞
Using (4.14) and Hölder’s inequality, we have, for any n ∈ N and any s ∈ [0, 1],
|u(tn + s) − ϑ|H ≤ |(˜a ∗ v)(tn + s)|H + |(˜a ∗ v)(tn )|H + a∞
tn + s
|v(τ )| dτ 2 H
1/2
tn
+ (a ∗ v)(tn ) − lim (a ∗ v)(tm ) . m→∞
Since v ∈ L2 (R+ ; H ) and (4.16) holds, we obtain u(tn + s) → ϑ as n → ∞ for all s ∈ [0, 1]. By continuity of E , this in turn implies that E (u(tn + s)) → E (ϑ) as n → ∞ for all s ∈ [0, 1]. Consequently 1
E (ϑ) = lim
n→∞
E (u(tn + s)) ds,
(4.17)
0
by the dominated convergence theorem. Now, integrating V0 (tn + ·) defined in (4.2) over [0, 1], we obtain 1
V (tn + s) ds =
1
0
E (u(tn + s)) ds + 0
tn +1
+
1
2
tn +1
(ζ ∗ |v|2H )(s) ds
tn
∞
−⟨f (s), (ζ ∗ v)(s)⟩H + M
tn
2 ˙ |f (τ )| + |f (τ )|H dτ ds, 2 H
s
which shows that 1
V0 (tn + s) ds = E (ϑ),
V∞ = lim
n→∞
(4.18)
0
by (4.17), (i), (Af′ ), and the simple estimate
tn + 1
tn
2 ⟨f (s), (ζ ∗ v)(s)⟩H ds ≤ |ζ |L1 (R+ )
tn + 1
|f (s)|2H ds
tn
tn + 1
(ζ ∗ |v|2H )(s) ds.
tn
Since ϑ was chosen arbitrarily in ω(u), (4.18) implies that E is constant on ω(u). In view of (i), (Af′ ), and the structure of V0 , we also see that limt →∞ E (u(t )) = V∞ . Hence (ii) is proven. Finally, to prove (iii), let again ϑ ∈ ω(u) and tn ↗ ∞ be such that limn→∞ u(tn ) = ϑ . Recall that we have already shown that u(tn + s) → ϑ as n → ∞ for all s ∈ [0, 1]. Therefore E ′ (u(tn + s)) → E ′ (ϑ) as n → ∞ for all s ∈ [0, 1]. Since E ′ (u) = −v + f , we have, by the dominated convergence theorem,
E ′ (ϑ) = lim
1
n→∞
E ′ (u(tn + s)) ds 0
tn + 1
= lim
n→∞
−v(s) + f (s) ds = 0,
tn
where the last step follows from the fact that v ∈ L2 (R+ ; H ), using (Af′ ) and Hölder’s inequality.
4.1. Proof of Theorem 4.1 Let u be a globally bounded solution of (4.1) and let ϑ ∈ ω(u). Setting V1 (t ) = V0 (t ) − E (ϑ), for t ≥ 0, where V0 is as in (4.2), we know that V1 is nonnegative, nonincreasing, and locally absolutely continuous on R+ . Further, limt →∞ V1 (t ) = 0. In addition,
γ 1 − V˙ 1 (t ) ≥ a∞ |v(t )|2H + (ζ ∗ |v|2H )(t ), 2
4
for a.a. t > 0.
(4.19)
All these properties follow from Proposition 4.3 and the proof of Proposition 4.4. If V1 (t0 ) = 0 for some t0 ≥ 0, then V1 (t ) = 0 for all t ≥ t0 . Therefore, we may assume that V1 (t ) is strictly positive on R+ .
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Using Young’s inequality and Lemma 4.2, we deduce from the definitions of V0 and V1 that V1 (t )1−θ ≤ C1
|E (u(t )) − E (ϑ)|1−θ + [(ζ ∗ |v|2H )(t )]
2(1−θ) 2
+ |f (t )|H + [(ζ ∗ |v|2H )(t )]
1−θ 2θ
∞
(|f (τ )|2H + |f˙ (τ )|2H ) dτ
+
2(12−θ)
t
,
t ≥ 0,
(4.20)
for some constant C1 > 0. Since θ ∈ (0, 1/2], we have 2(1 − θ ) ≥ 1 and (1 − θ )/θ ≥ 1. Using this and the fact that both (ζ ∗ |v|2H )(t ) and the integral in (4.20) tend to zero as t → ∞, we obtain
V1 ( t )
1−θ
≤ C2 |E (u(t )) − E (ϑ)|
1−θ
1 2
+ |f (t )|H + [(ζ ∗ |v| )(t )] + 2 H
∞
(|f (τ )| + |f˙ (τ )|2H ) dτ 2 H
12
,
t
t ≥ t∗ ,
(4.21)
where C2 > 0 is a constant and t∗ > 0 is sufficiently large. Next, we introduce the open set
Ωσ = {t ∈ (t∗ , ∞) : |u(t ) − ϑ|H < σ }. Restricting t in (4.21) to Ωσ , we may use the Łojasiewicz inequality for E near ϑ as well as the equality E ′ (u) = −v + f to get
V1 ( t )
1−θ
1 2
≤ C3 |v(t )|H + |f (t )|H + [(ζ ∗ |v| )(t )] + 2 H
∞
(|f (τ )| + |f˙ (τ )| ) dτ 2 H
2 H
12
,
t ∈ Ωσ ,
(4.22)
t
with some constant C3 > 0. From (4.19) and (4.22) we then obtain for a.a. t ∈ Ωσ that
−
d dt
[V1 (t )θ ] = −θ V1 (t )θ−1 V˙ 1 (t ) θ
2
a∞ |v(t )|2H +
γ
(ζ ∗ |v|2H )(t ) ≥ ∞ 1 1 C3 |v(t )|H + |f (t )|H + [(ζ ∗ |v|2H )(t )] 2 + t (|f (τ )|2H + |f˙ (τ )|2H ) dτ 2 1
≥ C4 |v(t )|H + [(ζ ∗ |v| )(t )] 2 H
1 2
4
− C5 |f (t )|H +
∞
(|f (τ )| + |f˙ (τ )| ) dτ 2 H
2 H
12
,
(4.23)
t
where C4 , C5 > 0 are constants. Now, integrating (4.23) over Ωσ and using (Af′ ) we obtain that v ∈ L1 (Ωσ ; H ). We now show that v ∈ L1 (R+ ; H ). To this end, choose tn ↗ ∞ such that limn→∞ u(tn ) = ϑ . We may assume that tn ∈ Ωσ for all n ∈ N. Define next exit times sn by means of sn = sup{t > tn : [tn , t ] ⊂ Ωσ }. Then there exists N ∈ N such that sN = ∞. If the contrary were true, we would have |u(sn ) − ϑ|H = σ > 0 for all n ∈ N. On the other hand, we get from (4.14) that
|u(sn ) − ϑ|H ≤ |(˜a ∗ v)(sn )|H + |(˜a ∗ v)(tn )|H + a∞
t
sn
n
|v(τ )|H dτ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) m→∞
H
≤ |(˜a ∗ v)(sn )|H + |(˜a ∗ v)(tn )|H + a∞ |v(τ )|H dτ (tn ,∞)∩ Ωσ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) → 0 as n → ∞, m→∞
H
due to (4.16) and the fact that v ∈ L1 (Ωσ ; H ). This is a contradiction, and therefore sN = ∞ for some N ∈ N. Hence v ∈ L1 (R+ ; H ), which together with (4.14) and (4.16) implies that limt →∞ u(t ) = ϑ in H. Thus our claim follows from the relative compactness of u in V . The proof is now complete. Acknowledgments The authors are grateful to the referee for his/her comments and suggestions. The first author was partially supported by FONDECYT Grant 1100559 and by Universidad de La Serena Proyecto DIULS CD091501. The second author was partially supported by FONDECYT Grant 1110033.
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Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Long-time behavior of nonlinear integro-differential evolution equations Justino Sánchez a , Vicente Vergara b,∗ a
Universidad de La Serena, Departamento de Matemáticas, Avenida Cisternas 1200, La Serena, Chile
b
Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta N. 1520, Arica, Chile
article
info
Article history: Received 21 January 2013 Accepted 12 June 2013 Communicated by S. Carl
abstract We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space. © 2013 Published by Elsevier Ltd
MSC: primary 37B25 37B35 secondary 45M05 35B40 Keywords: Integro-differential evolution equations Fractional derivative Gradient sytem Lyapunov function Convergence to steady state Lojasiewicz–Simon inequality
1. Introduction In this paper we study the long-time behavior of globally bounded strong solutions to nonlinear evolutionary equations with memory of the form d dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds + k∞ u(t ) + E ′ (u(t )) = f (t ), 0
t > 0,
(1.1)
u(0) = u0 , in a real Hilbert space H. Here E ′ is the Fréchet derivative of a functional E ∈ C 1 (V ), where V is a Hilbert space which densely and continuously injects into H. The vector-valued function f and the scalar kernel k1 are locally integrable functions on R+ ; k0 , and k∞ are real constants, and k0 is chosen nonnegative. The vector u0 ∈ V stands for the initial condition for the unknown function u. Under suitable conditions on k1 and f , we prove that any globally bounded solution of (1.1) converges to a steady state in V , that is, u∞ := limt →∞ u(t ) exists in V and u∞ is a solution of the problem k∞ u∞ + E ′ (u∞ ) = 0,
∗
Corresponding author. E-mail addresses: [email protected] (J. Sánchez), [email protected] (V. Vergara).
0362-546X/$ – see front matter © 2013 Published by Elsevier Ltd http://dx.doi.org/10.1016/j.na.2013.06.006
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
21
provided that the solution u has relatively compact range in V and the functional E satisfies the Łojasiewicz–Simon inequality (see Definition 3). Note that, without loss of generality, we may set k∞ = 0 in (1.1) since we can include it into the nonlinear term E by defining a new nonlinearity k∞
E˜ (u) =
2
|u|2H + E (u).
In the literature one finds many papers for problems of the form (1.1), as well as variants of them which are studied in a strong setting, assuming smoothness on the nonlinearities. See e.g. [1–5]. It seems that results about convergence to steady state of globally bounded solutions of (1.1) under the general setting given above are unknown, except for some special cases, where convergence results are well-known. Examples of the latter are the following. In case k0 = 1 and k1 = k∞ = 0, Eq. (1.1) becomes a first order problem of the form ut + E ′ ( u) = f ,
t > 0,
u(0) = u0 , which has been studied by many authors; see e.g. [6–8]. See also [9]. The case k0 = k∞ = 0 and k1 ̸= 0 of Eq. (1.1) becomes a problem of the form d
t
dt
k1 (t − s)(u − u0 )(s)ds
+ E ′ (u(t )) = f (t ),
t > 0,
(1.2)
0
u(0) = u0 , which has been studied by Vergara and Zacher [10] in a finite-dimensional Hilbert space. In this paper we extend the results obtained in [10] to real Hilbert spaces for Eq. (1.1). More precisely, we study integrodifferential evolution equations of two types. – Problems of order 1 (k0 > 0) with memory (k1 ̸= 0): d
dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds
+ E ′ (u(t )) = f (t ),
t > 0,
0
u(0) = u0 . – Problems of order less than 1 (k0 = 0): d
t
dt
k1 (t − s)(u(s) − u0 )ds
+ E ′ (u(t )) = f (t ),
t > 0,
0
u(0) = u0 . 1 ,2
Concerning applications, let Ω be a bounded domain in Rn with smooth boundary ∂ Ω . Setting H = L2 (Ω ), V = W0 (Ω ) (the classical Sobolev space) and
E (v) =
1
2
|∇v|2 dx +
Ω
Ω
v ∈ W01,2 (Ω ),
G(x, v)dx,
problem (1.1) becomes a problem of the form d dt
k0 u(t ) +
u = 0,
t
k1 (t − s)(u(s) − u0 )ds
+ k∞ u(t ) − ∆u(t ) + g (x, u(t )) = f (t ),
in R+ × Ω ,
0
on R+ × ∂ Ω ,
u(0) = u0 ,
in Ω ,
(1.3)
where g (x, s) := ∂s G(x, s) for all s ∈ R, which arises as a model for nonlinear heat flow in material with memory; see e.g., Gurtin and Pipkin [11], MacCamy [12], Nunziato [13], and the monograph Prüss [4]. See also [2], where existence and uniqueness results of globally bounded solutions of (1.3) were obtained for more general functions g. Concerning convergence to steady state for nonlinear equations with memory, there has been partial progress; see e.g. [14–20]. The reason for this lies essentially in the fact that these problems do not generate in general a semiflow in the natural phase space. Another technical difficulty consists in proving that the solutions of such problems are globally bounded and have relatively compact range in a natural phase space. Further, for problems with memory, it is in general a highly nontrivial task to construct Lyapunov functions (cf. [21, Chapter 14] and [17]) which are appropriate to investigating the asymptotic behavior of globally bounded solutions. The paper [10] develops a method for finding Lyapunov functions for problems of order less than 1, and for orders between 1 and 2 (in time), in finite-dimensional Hilbert spaces, which combined with the Łojasiewicz inequality allows one to prove convergence to a single steady state of bounded solutions of such problems.
22
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Recently, the second author [22] extended these results to real Hilbert spaces for problems of order between 1 and 2. The purpose of this paper is to deal with problems of order less than 1 and of order 1 with memory, in order to close this gap. In this article we show that any globally bounded strong solution of (1.1) (see Definition 2) converges to a single steady state. Here we make use of ideas and results obtained by Vergara and Zacher [10] and Vergara [22]. By means of a suitable approximation technique our convergence result can properly be formulated to any globally bounded weak solution. This will be treated in a separate work. The proof of our main results Theorems 3.1 and 4.1 concerning the convergence to the steady state is based on an appropriate new Lyapunov function, compactness properties, and on the Łojasiewicz–Simon inequality. The paper is organized as follows. Section 2 recalls some basic definitions and collects important properties of the operator Bv = dtd (k ∗ v) with nonnegative, nonincreasing kernel k, including the key result from [10]. In Section 3 we study problems of order 1 with memory and we state the main result Theorem 3.1 in this section. In Section 4 we discuss problems of order less than 1, in which we state the corresponding result Theorem 4.1. 2. Preliminaries Let V and H be real Hilbert spaces such that V is densely and continuously embedded into H. We shall identify H with its dual H ′ , that is, we have V ↩→ H ≈ H ′ ↩→ V ′ . We denote by ⟨·, ·⟩ the scalar products and duality relations; these will be made precise by subscripts according to the respective space. The operator E ′ is nonlinear and continuous from V into V ′ , and it is the Fréchet derivative of a functional E ∈ C 1 (V ). We will use the standard notation for the time derivative of a function u, that is, dtd u = u˙ = ut . For an interval J := [0, T ] (T > 0), s > 0 and 1 < p < ∞, we denote by Hps (J ; H ) the vector-valued Bessel potential space. We recall that if s ∈ N then Hps (J ; H ) stands for the classical Sobolev space. For s ∈ (0, 1], we denote by 0 Hps (J ; H ) the subspace of all functions f in Hps (J ; H ) with f (0) = 0 provided this trace exists; see e.g. [23]. The convolution of two functions a, b ∈ L1,loc (R+ ) (R+ := [0, ∞)) is defined by (a ∗ b)(t ) = t ≥ 0.
t 0
a(t − s)b(s)ds,
Lemma 2.1. Let H be a real Hilbert space and T > 0. Then for each k ∈ H11 ([0, T ]) and each v ∈ L2 ([0, T ]; H ) we have
d dt
(k ∗ v)(t ), v(t )
= H
1 d 2 dt
+
(k ∗ |v|2H )(t ) +
2
k(t )|v(t )|2H
t
1
1 2
[−k˙ (s)] |v(t ) − v(t − s)|2H ds,
for a.a. t ∈ (0, T ).
(2.1)
0
See [10, Lemma 2.2] for the proof. Next, we state a recent result due to Vergara and Zacher [10] which is the key to finding a proper Lyapunov function for Eq. (1.1). Theorem 2.2. Let H be a real Hilbert space, T > 0, and a ∈ L1,loc (R+ ) be nonnegative and nonincreasing such that k ∗ a = 1 in (0, ∞) for some nonnegative k ∈ L1,loc (R+ ). Suppose that v ∈ L2 ([0, T ]; H ) and that there exists x ∈ H such that a ∗ (v − x) ∈ 0 H21 ([0, T ]; H ), as well as a ∗ |v − x|2H ∈ 0 H11 ([0, T ]). Then
d dt
1 d 1 (a ∗ v)(t ), v(t ) ≥ (a ∗ |v|2H )(t ) + a(t )|v(t )|2H , H
2 dt
2
for a.a. t ∈ (0, T ).
(2.2)
The assumption a ∗ |v − x|2H ∈ 0 H11 ([0, T ]) in Theorem 2.2 seems a little awkward from the point of view of applications. We will now describe a rather wide class of kernels a for which we can guarantee that this assumption holds. ˆ In what follows ∞f stands for the Laplace transform of f . A function a ∈ L1,loc (R+ ) is said to be of subexponential growth if, for all ε > 0, 0 e−εt |a(t )| dt < ∞. Following [4, Definition 3.3] we say that a kernel a ∈ L1,loc (R+ ) of subexponential growth is 1-regular if there is a constant c > 0 such that |λˆa′ (λ)| ≤ c |ˆa(λ)| for all Reλ > 0. A kernel a ∈ L1,loc (R+ ) of subexponential growth satisfying aˆ (λ) ̸= 0, Re λ > 0, is called θ -sectorial (θ > 0) if | arg aˆ (λ)| ≤ θ for all Re λ > 0 (cp. [4, Definition 3.2]). The following class of kernels has been introduced in [24, Definition 2.6.3]. See also [25, Definition 2.1]. Definition 1. Let a ∈ L1,loc (R+ ) be of subexponential growth, and let θa > 0 and α ≥ 0. Then a is said to belong to the class K 1 (α, θa ) if a is 1-regular and θa -sectorial, and satisfies lim sup | aˆ (µ)| µα < ∞, µ→∞
lim inf | aˆ (µ)| µα > 0, µ→∞
We have now the following result.
and
lim inf | aˆ (µ)| > 0. µ→0
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
23
Proposition 2.3. Let H be a real Hilbert space, T > 0, and a ∈ L1,loc (R+ ) be such that k ∗ a = 1 in (0, ∞) for some k ∈ K 1 (α, θ ) with α ∈ (0, 1) and θ < π . Suppose that v ∈ L2 ([0, T ]; H ) and a ∗ v ∈ 0 H21 ([0, T ]; H ). Then we have a ∗ |v|2H ∈ 0 H11 ([0, T ]). For the proof of this proposition see [10, Proposition 2.1] and [25, Corollary 2.1]. Definition 2. A function u ∈ C (J ; V ) is called (a) a strong solution of (1.1) if u ∈ H21 (J ; H ) ∩ C (J ; V ) and (1.1) holds a.e. on J; (b) a globally bounded strong solution of (1.1) if u is a strong solution on each interval J and u ∈ L∞ (R+ ; V ). Henceforth we simply write ‘‘solution’’ instead of ‘‘strong solution’’. Definition 3. We say that a functional E ∈ C 1 (V ) satisfies the Łojasiewicz–Simon inequality in the neighborhood of some point ϑ ∈ V if there exist constants θ ∈ (0, 1/2], C > 0, and σ > 0 such that, for all u ∈ V with |u − ϑ|V ≤ σ , we have
|E (u) − E (ϑ)|1−θ ≤ C |E ′ (u)|V ′ . Here the constant θ is called the Łojasiewicz exponent. A typical example of a functional E is given by
E (u) =
1 2
η(u, u) +
Ω
ϕ(x, u)dx,
u ∈ V,
where η : V × V → R is a bilinear, continuous, symmetric and coercive form, and the nonlinear term ϕ(x, ·) belongs to C 2 (V ) for a.a. x ∈ Ω . We refer the reader to [6,8] for a comprehensive study of this subject. For every globally bounded solution u of (1.1) we define the ω-limit set by
ω(u) = {ϑ ∈ V : there exists (tn ) ↗ ∞ s.t. u(tn ) → ϑ in V } . 3. Problems of order 1 with memory In this section we investigate problems of the form d dt
k0 u(t ) +
t
k1 (t − s)(u(s) − u0 )ds + E ′ (u(t )) = f (t ),
t > 0,
u(0) = u0 ,
(3.1)
0
where k0 is a positive constant and the following assumptions are satisfied. Throughout the paper we set v := u0 )).
d dt
(k1 ∗ (u −
(A1) k1 ∈ L1,loc (R+ ) is nonnegative and nonincreasing. (A2) There is a nonnegative nonincreasing kernel a ∈ L1,loc (R+ ) such that
(k1 ∗ a)(t ) = 1,
t > 0.
(A3) There exists a constant γ > 0 such that the solution ζ of
ζ (t ) + γ
t
ζ (s)ds = a(t ),
t > 0,
(3.2)
0
is nonnegative. (A4) f ∈ L1 (R+ ; H ) ∩ L2 (R+ ; H ) is such that the function h(t ) :=
∞
∞
t
|f (τ )|2H dτ
21 ds
s
is bounded on R+ .
(AE ) the functional E : V → R belongs to C 1 (V ). Remark 1. (a) Typical examples of kernels k1 and a, which satisfy (A1)–(A3) are given by k1 (t ) = g1−α (t )e−γ t ,
(3.3)
and a(t ) = gα (t )e−ωt + γ (1 ∗ [gα e−γ · ])(t ),
t > 0, α ∈ (0, 1), γ > 0,
24
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
respectively, where gβ (t ) = t β−1 /Γ (β), t > 0, β > 0, and Γ (·) is the Gamma function. Furthermore, we have that a ∈ K 1 (α, (1 + α)π /2) and k1 ∈ K 1 (α, απ /2); see [24, Example 2.1]. (b) For each γ > 0 the unique solution of (3.2) is given by
ζγ (t ) := a(t ) − γ (e−γ · ∗ a)(t ),
t > 0.
Hence, if condition (A3) holds, by decreasing γ , we may assume that ζγ is strictly positive and strictly decreasing on (0, ∞). ∞ Furthermore, ζ ∈ L1 (R+ ) and limt →∞ ζ (t ) = 0, therefore a∞ := limt →∞ a(t ) = γ 0 ζ (s)ds > 0; see [10, Remark 3.1]. (c) Assumptions (A1)–(A3) imply that k1 ∈ L1 (R+ ) with |k1 |L1 ≤ 1/a∞ . (d) Assumption (Af) was first introduced by Huang and Takáč in [9, Remark 3.1] to prove convergence to equilibrium of bounded solutions for a nonlinear heat equation with a time-dependent perturbation f . (e) Under the condition (Af) the function h belongs to C01 (R+ ). In particular, if f has polynomial or exponential decay to zero as time tends to infinity then h has the same property. (f) Let u be a solution of (3.1). Then a ∗v = u − u0 ∈ 0 H21 (J ; H ) by assumption (A2) and v ∈ L2 (J ; H ) by (Af)–(AE ) and (3.1). Hence if we assume that a ∈ K 1 (α, θa ), we obtain by Proposition 2.3 that a ∗ |v|2H ∈ 0 H11 (J ). Thus we can apply inequality (2.2) in Theorem 2.2 to the function v . Now we establish our first main result on convergence of globally bounded solutions of (3.1). Theorem 3.1. Let u be a globally bounded solution of (3.1) with k0 > 0, u0 ∈ V , and assume that a ∗ |v|2H ∈ 0 H11,loc (R+ ). Suppose that (i) the set {u(t ) : t ≥ 0} is relatively compact in V ; (ii) (A1)–(A3), (Af), and (AE ) hold; (iii) E satisfies the Łojasiewicz–Simon inequality near each point ϑ ∈ ω(u) ⊂ V . Then limt →∞ u(t ) = ϑ in V , and ϑ is a stationary solution, i.e. E ′ (ϑ) = 0. Before beginning the proof of Theorem 3.1, we state some auxiliary facts about properties of the ω-limit set ω(u) as well as of u itself. The proof of the following result shows how one may construct a proper Lyapunov function for Eq. (3.1). Proposition 3.2. Let u be a globally bounded solution of Eq. (3.1) and let ϵ ∈ (0, 2k0 ). Assume that (A1)–(A3), (Af), and a ∗ |v|2H ∈ 0 H11,loc (R+ ) hold. Then the function V0 : R+ → R defined by V0 ( t ) =
1 2
(ζ ∗ |v| )(t ) + E (u(t )) + 2 H
1
2ϵ
∞
|f (s)|2H ds,
t > 0,
(3.4)
t
is locally absolutely continuous and nonincreasing on R+ . Furthermore,
−
d dt
V0 (t ) ≥ k0 −
ϵ 2
|ut |2H +
γ 2
ζ ∗ |v|2H +
1 2
a∞ |v(t )|2H ≥ 0,
for a.a. t > 0.
(3.5)
Proof. We first multiply (3.1) by ut and obtain k0 |ut |2H + ⟨v, ut ⟩H +
d dt
E (u) = ⟨f , ut ⟩H .
(3.6)
Next we estimate separately the terms ⟨v, ut ⟩H and ⟨f , ut ⟩H . Note that from (A2) we get
⟨v, ut ⟩H =
d dt
k1 ∗ (u − u0 ), ut
= H
d dt
a ∗ v, v
,
(3.7)
H
and by Theorem 2.2 and (A3) we have
d dt
a ∗ v, v
≥ H
≥
1 d 2 dt 1 d 2 dt
a ∗ |v|2H +
ζ ∗ |v|2H +
1 2
γ 2
a(t )|v(t )|2H
ζ ∗ |v|2H +
1 2
a∞ |v(t )|2H .
(3.8)
Next we simply estimate the term ⟨f , ut ⟩H by Young’s inequality, i.e.
⟨f , ut ⟩H ≤
1 2ϵ
ϵ |f |2H + |ut |2H .
(3.9)
2
Replacing (3.7)–(3.9) by (3.6) we obtain (3.5) with V0 defined by (3.4).
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
25
Proposition 3.3. Let u be a globally bounded solution of Eq. (3.1). Assume that (A1)–(A3), (Af)–(AE ) hold, a ∗ |v|2H ∈ 1 0 H1,loc (R+ ), and the set {u(t ) : t ≥ 0} is relatively compact in V . Then: (i) ut , v ∈ L2 (R+ ; H ) and ζ ∗ |v|2H ∈ L1 (R+ ) ∩ C0 (R+ ); (ii) the potential E is constant on ω(u) and limt →∞ E (u(t )) exists; (iii) for every ϑ ∈ ω(u) we have E ′ (ϑ) = 0. Proof. Since V0 defined by (3.4) is absolutely continuous and decreasing on R+ and E (u) is bounded from below, we have that V0∞ := limt →∞ V0 (t ) exists. Thus integrating (3.5) we prove (i) except that ζ ∗ |v|2H ∈ C0 (R+ ). This will be shown later. Since u has relatively compact range in V , it follows that the ω(u) is nonempty, compact and connected. Let ϑ ∈ ω(u) and choose tn ↗ ∞ such that u(tn ) → ϑ in V . Since ut ∈ L2 (R+ ; H ), we obtain u(tn + s) = u(tn ) +
tn + s
ut (τ )dτ → ϑ
in H , for every s ∈ [0, 1].
tn
This, together with the relative compactness of the trajectory, implies that u(tn + s) → ϑ in V for every s ∈ [0, 1]. Therefore, limn→∞ E (u(tn + s)) = E (ϑ) for every s ∈ [0, 1], and thus, by the dominated convergence theorem, 1
E (ϑ) = lim
n→∞
E (u(tn + s))ds. 0
In addition, integrating V0 (tn + ·) defined in (3.4) over [0, 1], we obtain
E (ϑ) + lim
n→∞
tn + 1
1 2
tn
(ζ ∗ |v| )(s) + 2 H
1 2ϵ
∞
|f (τ )|
2 Hd
n→∞
s
t
+1
1
τ ds = lim ∞
V0 (tn + s)ds = V0∞ . 0
1 Now, from assumption (Af) we obtain that limn→∞ t n |f (τ )|2H dτ ds = 0, and since ζ ∗ |v|2H ∈ L1 (R+ ) we get 2ϵ s n E (ϑ) = V0∞ , that is, E is constant on ω(u). Further, as a consequence of the above, we obtain that (ζ ∗ |v|2H )(t ) → 0 as t → ∞. Indeed, if the opposite were true then there would exist a δ > 0 and a sequence tn → ∞ as n → ∞ such that (ζ ∗ |v|2H )(tn ) ≥ δ for all n ∈ N. By compactness, there exists a subsequence tnk such that E (u(tnk )) → V0∞ as k → ∞, hence (ζ ∗ |v|2H )(tnk ) → 0 as k → ∞, which is a contradiction. Hence (ζ ∗ |v|2H )(t ) → 0 as t → ∞. Moreover, we see that limt →∞ E (u(t )) = V0∞ . Hence the claim (ii) is proved and the proof of (i) is complete. Next, since E ∈ C 1 (V ), we have E ′ (u(tn +s)) → E ′ (ϑ) in V ′ for every s ∈ [0, 1]. Further, using the dominated convergence theorem, Eq. (3.1), (i), and assumption (Af) we obtain
E ′ (ϑ) = lim
n→∞
tn + 1
E ′ (u(s))ds
tn
= lim − n→∞
tn + 1
(k0 ut (s) + v(s)) ds +
tn
tn + 1
f (s)ds = 0.
tn
The proof of the proposition is now complete.
3.1. Proof of Theorem 3.1 Let ϑ ∈ ω(u) and define a new energy function V1 : R+ → R by V1 (t ) := V0 (t ) − E (ϑ),
t > 0.
(3.10)
Thus V1 is nonnegative, nonincreasing, locally absolutely continuous on R+ , and limt →∞ V1 (t ) = 0. In addition, we may assume that V1 (t ) > 0 for all t > 0. This is because, if there exists a t > 0 such that V1 (t ) = 0, then V1 (s) = 0 for all s ≥ t and in this case u(t ) = ϑ for all t > 0. Now we will use our main assumption (iii). Let ϑ ∈ ω(u), since E is constant on ω(u), it follows from assumption (iii) and compactness of the ω-limit set that there exists an open set U ⊂ V such that ω(u) ⊂ U, and there are constants θ ∈ (0, 1/2] and C > 0 such that the inequality
|E (u(t )) − E (ϑ)|1−θ ≤ C |E ′ (u(t ))|V ′
(3.11)
holds for every u(t ) ∈ U. Further, since limt →∞ dist(u(t ), ω(u)) = 0, we see that there exists t ∗ ≥ 0 such that u(t ) ∈ U for all t ≥ t ∗ and (3.11) holds. Next, we compute and estimate the time derivative of V1 (t )θ . By (3.10) we obtain
V1 (t )
1−θ
≤ C1 |E (u(t )) − E (ϑ)|
1−θ
+ (ζ ∗ |v| )
2 1−θ H
∞
|f (s)|
1−θ
2 H ds
+ t
2 21 2(1−θ) H
≤ C2 |E (u(t ))|V ′ + (ζ ∗ |v| ) ′
∞
|f (s)|
21 2(1−θ )
2 H ds
+ t
.
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Since, 2(1 − θ ) ≥ 1 and (1 − θ )/θ ≥ 1 for all θ ∈ (0, 1/2], we conclude that
V1 ( t )
1−θ
2 21 H
≤ C3 |E (u(t ))|V ′ + (ζ ∗ |v| ) + ′
∞
|f (s)|
2 H ds
21
.
(3.12)
t
Therefore, from (3.5) and (3.12), we have
−
d dt
d V0 (t ) dt θ C0 |ut |2H + ζ ∗ |v|2H + |v|2H
[V1 (t )θ ] = −θ V1 (t )θ−1 ≥
12 2 | f ( s )| ds H t 2 θ C0 |ut |H + ζ ∗ |v|2H + |v|2H ≥ ∞ 1 1 C˜ 4 |ut |H + (ζ ∗ |v|2H ) 2 + |v|H + |f (t )|H + t |f (s)|2H ds 2 1
C4 |E ′ (u(t ))|V ′ + (ζ ∗ |v|2H ) 2 +
2 21 H
≥ C5 |ut |H + (ζ ∗ |v| ) + |v|H
∞
˜ − C5 |f (t )|H +
∞
|f (s)|
21
2 H ds
.
(3.13)
t
This in turn implies that ut ∈ L1 ([t ∗ , ∞), H ). Therefore limt →∞ u(t ) exists in H. Hence, from the relative compactness of u(t ) in V , our claim follows. 4. Problems of order less than 1 In this section we investigate problems of the form d dt
t
k1 (t − s)(u(s) − u0 )ds + E ′ (u(t )) = f (t ),
t > 0,
u(0) = u0 .
(4.1)
0
We will assume that the following assumptions hold. (A1) k1 ∈ L1,loc (R+ ) is nonnegative and nonincreasing. (A2) There is a nonnegative nonincreasing kernel a ∈ L1,loc (R+ ) such that
(k1 ∗ a)(t ) = 1,
t > 0.
(A3) There exists a constant γ > 0 such that the solution ζ of
ζ (t ) + γ
t
ζ (s)ds = a(t ),
t > 0,
0
is nonnegative.
(Af′ ) f ∈ H11 (R+ ; H ) ∩ H21 (R+ ; H ) is such that the function 21 ∞ ∞ ds h˜ (t ) := |f (τ )|2H + |f˙ (τ )|2H dτ t
s
is bounded on R+ . (AE ) the functional E : V → R belongs to C 1 (V ). Remark 2. (a) Assumptions (A1)–(A3) and (AE ) are the same as above, while (Af′ ) is clearly different from (Af). This new condition (Af′ ) was first used by Vergara and Zacher [10] to get convergence results for (4.1) in finite dimensional spaces. (b) Under the condition (Af′ ) the function h˜ belongs to C01 (R+ ). In particular, if f and f˙ have polynomial decay or exponential decay to zero as time tends to infinity then h˜ has the same property. t (c) Let u be a solution of (4.1) and set v(t ) := 0 k1 (t − s)(u(s) − u0 )ds as in the above section. Then a ∗ v = u − u0 ∈ 1 ′ 1 0 H2 (J ; H ) by assumption (A2) and v belongs to L2 (J ; H ) by (Af )–(AE ) and (4.1). Hence if we assume that a ∈ K (α, θa ), we 2 1 obtain by Proposition 2.3 that a ∗ |v|H ∈ 0 H1 (J ). Thus we can apply inequality (2.2) in Theorem 2.2 to the function v . (d) Notes (a)–(c) in Remark 1 still hold. Now we establish our main result on convergence of globally bounded solutions of (4.1). Theorem 4.1. Let u be a globally bounded solution of (4.1), u0 ∈ V , and assume that a ∗ |v|2H ∈ 0 H11,loc (R+ ). Suppose that (i) the set {u(t ) : t ≥ 0} is relatively compact in V ;
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
27
(ii) (A1)–(A3), (Af′ ), and (AE ) hold; (iii) E satisfies the Łojasiewicz–Simon inequality near each point ϑ ∈ ω(u) ⊂ V . Then limt →∞ u(t ) = ϑ in V , and ϑ is a stationary solution, i.e. E ′ (ϑ) = 0. Before beginning the proof of Theorem 4.1, we state some auxiliary facts about properties of the ω-limit set ω(u) as well as of u itself. It uses the following elementary lemma which will be used in the sequel (for the proof of this lemma, see for instance [10]). Lemma 4.2. Let H be a real Hilbert space and T > 0. Suppose that k ∈ L1,loc (R+ ) is nonnegative. Then for any v ∈ L2 ([0, T ]; H ) we have
|(k ∗ v)(t )|2H ≤ (k ∗ |v|2H )(t ) (1 ∗ k)(t ),
for a.a. t ∈ (0, T ).
The proof of the following result shows how one may construct a Lyapunov function for Eq. (4.1). Proposition 4.3. Let u be a globally bounded solution of Eq. (3.1). Assume that (A1)–(A3), (Af′ ) –(AE ) hold, and that a ∗ |v|2H ∈ 1 −1 }. Then the function V0 : R+ → R defined by 0 H1,loc (R+ ). Let M = 2|ζ |L1 (R+ ) max{γ , γ V0 (t ) =
1 2
(ζ ∗ |v|2H )(t ) + E (u(t )) − ⟨f (t ), (ζ ∗ v)(t )⟩H + M
∞
|f (τ )|2H + |f˙ (τ )|2H dτ ,
t ≥ 0,
(4.2)
t
is absolutely continuous and decreasing on R+ . Furthermore,
−
d dt
1
V0 (t ) ≥
2
a∞ |v(t )|2H +
γ 4
(ζ ∗ |v|2H )(t ) ≥ 0,
for a.a. t > 0.
(4.3)
Proof. We take the inner product of the integro-differential equation in (4.1) with ut to find that
⟨v, ut ⟩H +
d dt
E (u) = ⟨f , ut ⟩H ,
t > 0.
(4.4)
Hence, by (A2) we may write d
ut =
dt
( u − u0 ) =
d2 dt 2
[k1 ∗ a ∗ (u − u0 )] =
d dt
(a ∗ v),
(4.5)
which together with (4.4) yields
v,
d dt
d (a ∗ v) + E (u) = ⟨f , ut ⟩H , H
dt
t > 0.
(4.6)
Since v = −E ′ (u) + f ∈ L2,loc (R+ , H ) by the assumptions (AE ) and (Af′ ), a ∗ v = u − u0 ∈ 0 H21,loc (R+ ; H ) by the definition of a solution. Hence, we may apply inequality (2.2) to the first term in (4.6) to get the result d 1 dt
2
1 (a ∗ |v|2H )(t ) + E (u(t )) ≤ − a(t )|v(t )|2H + ⟨f (t ), ut (t )⟩H , 2
t > 0.
(4.7)
Using assumption (A3) we obtain from (4.7) that d dt
1 2
1 γ (ζ ∗ |v|2H )(t ) + E (u(t )) ≤ − a(t )|v(t )|2H − (ζ ∗ |v|2H )(t ) + ⟨f (t ), ut (t )⟩H , 2
2
t > 0.
(4.8)
In order to treat the term ⟨f , ut ⟩H , note that by Lemma 4.2 and Young’s inequality we have
⟨f , ut ⟩H = f , = ≤ ≤
d dt d dt d dt
d dt
(a ∗ v)
= f, H
d dt
(ζ ∗ v) + γ ⟨f , ζ ∗ v⟩H H
⟨f , ζ ∗ v⟩H − ⟨f˙ , ζ ∗ v⟩H + γ ⟨f , ζ ∗ v⟩H ⟨f , ζ ∗ v⟩H + 2|ζ |L1 (R+ ) γ |f |2H + γ −1 |f˙ |2H + γ ⟨f , ζ ∗ v⟩H + M |f |2H + |f˙ |2H + ζ ∗ |v|2H , 4
γ 4|ζ |L1 (R+ ) t > 0.
|ζ ∗ v|2H (4.9)
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J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
Hence, we obtain from (4.8) and (4.9) that the estimate
−
d dt
V0 (t ) ≥
1 2
holds in the a.e. sense.
a∞ |v(t )|2H +
γ 4
(ζ ∗ |v|2H )(t ),
t >0
(4.10)
We consider now globally bounded solutions of (4.1) with relative compact orbit in V . For any such solution u, the ω-limit set ω(u) is nonempty, compact and connected. Proposition 4.4. Let u be a globally bounded solution of (4.1). Suppose that (A1)–(A3), (AE )–(Af′ ) hold, and that a ∗ |v|2H ∈ 1 0 H1,loc (R+ ). Then: (i) v ∈ L2 (R+ ; H ) and ζ ∗ |v|2H ∈ L1 (R+ ) ∩ C0 (R+ ); (ii) the potential E is constant on ω(u) and limt →∞ E (u(t )) exists; (iii) for every ϑ ∈ ω(u) we have E ′ (ϑ) = 0. Proof. Let u be a globally bounded solution of (4.1). Clearly, E (u) is bounded from below. Furthermore, by Young’s inequality and Lemma 4.2,
⟨f (t ), (ζ ∗ v)(t )⟩H ≤ |ζ |L1 (R+ ) |f (t )|2H +
1 4
(ζ ∗ |v|2H )(t ),
t ≥ 0,
and so it is clear that V0 : R+ → R defined in (4.2) is bounded from below. By Proposition 4.3, the function V0 is nonincreasing and therefore limt →∞ V0 (t ) = inft ≥0 V0 (t ) =: V∞ exists. The first part of assertion (i) follows then directly from estimate (4.10). In order to see that ζ ∗ |v|2H ∈ C0 (R+ ), note first that |v|L∞ (R+ ;H ) ≤ C for some constant C > 0. In fact, this follows from the equality v = −E ′ (u) + f and the boundedness of u and f . Now, given ε > 0 we can choose δ > 0 so that |ζ |L1 (0,δ) C 2 ≤ ε . Define the function ζδ by ζδ (t ) = ζ (t ), t ∈ (0, δ), and ζδ (t ) = 0, t ≥ δ . Using Young’s inequality we may then estimate
(ζ ∗ |v|2H )(t ) ≤ |ζδ ∗ |v |2H |L∞ (R+ ) + ((ζ − ζδ ) ∗ |v|2H )(t ) ≤ ε + ((ζ − ζδ ) ∗ |v|2H )(t ),
t ≥ 0.
(4.11)
Since 0 ≤ (ζ − ζδ )(t ) ≤ ζ (δ) for a.a. t > 0, we further have
|((ζ − ζδ ) ∗ |v |2H )(t )| =
t /2
(ζ − ζδ )(s)|v(t − s)|2H ds +
0
≤ ζ (δ)
t
t /2
|v(s)|2H ds + C 2
t t /2
(ζ − ζδ )(s)|v(t − s)|2H ds
t
t /2
(ζ − ζδ )(s) ds,
which together with |v(·)|2H ∈ L1 (R+ ) and ζ ∈ L1 (R+ ) shows that limt →∞ ((ζ − ζδ ) ∗ |v|2H )(t ) = 0. The latter property, together with (4.11) and the fact that ε can be chosen arbitrarily small, imply that limt →∞ (ζ ∗ |v|2H )(t ) = 0. Hence (i) is established. Now let ϑ ∈ ω(u) and tn ↗ ∞ be such that limn→∞ u(tn ) = ϑ in V . Since u(t ) − u0 = (a ∗ v)(t ),
t ≥ 0,
(4.12)
cf. (4.5), it follows that limm→∞ (a ∗ v)(tm ) exists and that
ϑ = u0 + lim (a ∗ v)(tm ).
(4.13)
m→∞
Let again a∞ := limt →∞ a(t ) > 0 and a˜ := a − a∞ . Using (4.12) and (4.13) we have, for tn ≤ t with n ∈ N, u(t ) − ϑ = (a ∗ v)(t ) − lim (a ∗ v)(tm ) m→∞
= (a ∗ v)(t ) − (a ∗ v)(tn ) + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) m→∞ t = (˜a ∗ v)(t ) − (˜a ∗ v)(tn ) + a∞ v(τ ) dτ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) . tn
Observe that 0 ≤ a˜ (t ) = ζ (t ) − γ
∞
ζ (τ ) dτ ≤ ζ (t ), t
t > 0,
m→∞
(4.14)
J. Sánchez, V. Vergara / Nonlinear Analysis 91 (2013) 20–31
29
and thus, by Lemma 4.2, 1/2
|(˜a ∗ v)(t )|H ≤ [(ζ ∗ |v|2H )(t )]1/2 |ζ |L1 (R+ ) .
(4.15)
From (4.15) and (i) we deduce that lim (˜a ∗ v)(t ) = 0.
(4.16)
t →∞
Using (4.14) and Hölder’s inequality, we have, for any n ∈ N and any s ∈ [0, 1],
|u(tn + s) − ϑ|H ≤ |(˜a ∗ v)(tn + s)|H + |(˜a ∗ v)(tn )|H + a∞
tn + s
|v(τ )| dτ 2 H
1/2
tn
+ (a ∗ v)(tn ) − lim (a ∗ v)(tm ) . m→∞
Since v ∈ L2 (R+ ; H ) and (4.16) holds, we obtain u(tn + s) → ϑ as n → ∞ for all s ∈ [0, 1]. By continuity of E , this in turn implies that E (u(tn + s)) → E (ϑ) as n → ∞ for all s ∈ [0, 1]. Consequently 1
E (ϑ) = lim
n→∞
E (u(tn + s)) ds,
(4.17)
0
by the dominated convergence theorem. Now, integrating V0 (tn + ·) defined in (4.2) over [0, 1], we obtain 1
V (tn + s) ds =
1
0
E (u(tn + s)) ds + 0
tn +1
+
1
2
tn +1
(ζ ∗ |v|2H )(s) ds
tn
∞
−⟨f (s), (ζ ∗ v)(s)⟩H + M
tn
2 ˙ |f (τ )| + |f (τ )|H dτ ds, 2 H
s
which shows that 1
V0 (tn + s) ds = E (ϑ),
V∞ = lim
n→∞
(4.18)
0
by (4.17), (i), (Af′ ), and the simple estimate
tn + 1
tn
2 ⟨f (s), (ζ ∗ v)(s)⟩H ds ≤ |ζ |L1 (R+ )
tn + 1
|f (s)|2H ds
tn
tn + 1
(ζ ∗ |v|2H )(s) ds.
tn
Since ϑ was chosen arbitrarily in ω(u), (4.18) implies that E is constant on ω(u). In view of (i), (Af′ ), and the structure of V0 , we also see that limt →∞ E (u(t )) = V∞ . Hence (ii) is proven. Finally, to prove (iii), let again ϑ ∈ ω(u) and tn ↗ ∞ be such that limn→∞ u(tn ) = ϑ . Recall that we have already shown that u(tn + s) → ϑ as n → ∞ for all s ∈ [0, 1]. Therefore E ′ (u(tn + s)) → E ′ (ϑ) as n → ∞ for all s ∈ [0, 1]. Since E ′ (u) = −v + f , we have, by the dominated convergence theorem,
E ′ (ϑ) = lim
1
n→∞
E ′ (u(tn + s)) ds 0
tn + 1
= lim
n→∞
−v(s) + f (s) ds = 0,
tn
where the last step follows from the fact that v ∈ L2 (R+ ; H ), using (Af′ ) and Hölder’s inequality.
4.1. Proof of Theorem 4.1 Let u be a globally bounded solution of (4.1) and let ϑ ∈ ω(u). Setting V1 (t ) = V0 (t ) − E (ϑ), for t ≥ 0, where V0 is as in (4.2), we know that V1 is nonnegative, nonincreasing, and locally absolutely continuous on R+ . Further, limt →∞ V1 (t ) = 0. In addition,
γ 1 − V˙ 1 (t ) ≥ a∞ |v(t )|2H + (ζ ∗ |v|2H )(t ), 2
4
for a.a. t > 0.
(4.19)
All these properties follow from Proposition 4.3 and the proof of Proposition 4.4. If V1 (t0 ) = 0 for some t0 ≥ 0, then V1 (t ) = 0 for all t ≥ t0 . Therefore, we may assume that V1 (t ) is strictly positive on R+ .
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Using Young’s inequality and Lemma 4.2, we deduce from the definitions of V0 and V1 that V1 (t )1−θ ≤ C1
|E (u(t )) − E (ϑ)|1−θ + [(ζ ∗ |v|2H )(t )]
2(1−θ) 2
+ |f (t )|H + [(ζ ∗ |v|2H )(t )]
1−θ 2θ
∞
(|f (τ )|2H + |f˙ (τ )|2H ) dτ
+
2(12−θ)
t
,
t ≥ 0,
(4.20)
for some constant C1 > 0. Since θ ∈ (0, 1/2], we have 2(1 − θ ) ≥ 1 and (1 − θ )/θ ≥ 1. Using this and the fact that both (ζ ∗ |v|2H )(t ) and the integral in (4.20) tend to zero as t → ∞, we obtain
V1 ( t )
1−θ
≤ C2 |E (u(t )) − E (ϑ)|
1−θ
1 2
+ |f (t )|H + [(ζ ∗ |v| )(t )] + 2 H
∞
(|f (τ )| + |f˙ (τ )|2H ) dτ 2 H
12
,
t
t ≥ t∗ ,
(4.21)
where C2 > 0 is a constant and t∗ > 0 is sufficiently large. Next, we introduce the open set
Ωσ = {t ∈ (t∗ , ∞) : |u(t ) − ϑ|H < σ }. Restricting t in (4.21) to Ωσ , we may use the Łojasiewicz inequality for E near ϑ as well as the equality E ′ (u) = −v + f to get
V1 ( t )
1−θ
1 2
≤ C3 |v(t )|H + |f (t )|H + [(ζ ∗ |v| )(t )] + 2 H
∞
(|f (τ )| + |f˙ (τ )| ) dτ 2 H
2 H
12
,
t ∈ Ωσ ,
(4.22)
t
with some constant C3 > 0. From (4.19) and (4.22) we then obtain for a.a. t ∈ Ωσ that
−
d dt
[V1 (t )θ ] = −θ V1 (t )θ−1 V˙ 1 (t ) θ
2
a∞ |v(t )|2H +
γ
(ζ ∗ |v|2H )(t ) ≥ ∞ 1 1 C3 |v(t )|H + |f (t )|H + [(ζ ∗ |v|2H )(t )] 2 + t (|f (τ )|2H + |f˙ (τ )|2H ) dτ 2 1
≥ C4 |v(t )|H + [(ζ ∗ |v| )(t )] 2 H
1 2
4
− C5 |f (t )|H +
∞
(|f (τ )| + |f˙ (τ )| ) dτ 2 H
2 H
12
,
(4.23)
t
where C4 , C5 > 0 are constants. Now, integrating (4.23) over Ωσ and using (Af′ ) we obtain that v ∈ L1 (Ωσ ; H ). We now show that v ∈ L1 (R+ ; H ). To this end, choose tn ↗ ∞ such that limn→∞ u(tn ) = ϑ . We may assume that tn ∈ Ωσ for all n ∈ N. Define next exit times sn by means of sn = sup{t > tn : [tn , t ] ⊂ Ωσ }. Then there exists N ∈ N such that sN = ∞. If the contrary were true, we would have |u(sn ) − ϑ|H = σ > 0 for all n ∈ N. On the other hand, we get from (4.14) that
|u(sn ) − ϑ|H ≤ |(˜a ∗ v)(sn )|H + |(˜a ∗ v)(tn )|H + a∞
t
sn
n
|v(τ )|H dτ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) m→∞
H
≤ |(˜a ∗ v)(sn )|H + |(˜a ∗ v)(tn )|H + a∞ |v(τ )|H dτ (tn ,∞)∩ Ωσ + (a ∗ v)(tn ) − lim (a ∗ v)(tm ) → 0 as n → ∞, m→∞
H
due to (4.16) and the fact that v ∈ L1 (Ωσ ; H ). This is a contradiction, and therefore sN = ∞ for some N ∈ N. Hence v ∈ L1 (R+ ; H ), which together with (4.14) and (4.16) implies that limt →∞ u(t ) = ϑ in H. Thus our claim follows from the relative compactness of u in V . The proof is now complete. Acknowledgments The authors are grateful to the referee for his/her comments and suggestions. The first author was partially supported by FONDECYT Grant 1100559 and by Universidad de La Serena Proyecto DIULS CD091501. The second author was partially supported by FONDECYT Grant 1110033.
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