Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 47, pp. 1–35.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ROBUST EXPONENTIAL ATTRACTORS FOR
COLEMAN-GURTIN EQUATIONS WITH DYNAMIC BOUNDARY
CONDITIONS POSSESSING MEMORY
JOSEPH L. SHOMBERG
Abstract. Well-posedness of generalized Coleman-Gurtin equations equipped
with dynamic boundary conditions with memory was recently established by
the author with C. G. Gal. In this article we report advances concerning the
asymptotic behavior and stability of this heat transfer model. For the model
under consideration, we obtain a family of exponential attractors that is robust/Hölder continuous with respect to a perturbation parameter occurring in
a singularly perturbed memory kernel. We show that the basin of attraction of
these exponential attractors is the entire phase space. The existence of (finite
dimensional) global attractors follows. The results are obtained by assuming
the nonlinear terms defined on the interior of the domain and on the boundary satisfy standard dissipation assumptions. Also, we work under a crucial
assumption that dictates the memory response in the interior of the domain
matches that on the boundary.
1. Introduction to the model problem
In the framework of [23], let us only consider a thermodynamic process based
on heat conduction. Suppose that a bounded domain Ω ⊂ Rn , n ≥ 1, is occupied
by a body which may be inhomogeneous, but has a configuration constant in time.
Thermodynamic processes taking place inside Ω, with sources also present at the
boundary Γ, give rise to the following model for the temperature field u:
Z ∞
∂t u − ω∆u − (1 − ω)
k(s)∆u(x, t − s) ds + f (u)
0
(1.1)
Z ∞
+ α(1 − ω)
k(s)u(x, t − s) ds = 0,
0
in Ω × (0, ∞), subject to the boundary condition
Z ∞
∂t u − ω∆Γ u + ω∂n u + (1 − ω)
k(s)∂n u(x, t − s) ds
0
Z ∞
+ (1 − ω)
k(s)(−∆Γ + β)u(x, t − s) ds + g(u) = 0,
0
2010 Mathematics Subject Classification. 35B40, 35B41, 45K05, 35Q79.
Key words and phrases. Coleman-Gurtin equation; dynamic boundary conditions;
memory relaxation; exponential attractor; basin of attraction; global attractor;
finite dimensional dynamics; robustness.
c
2016
Texas State University.
Submitted August 15, 2015. Published February 10, 2016.
1
(1.2)
2
J. L. SHOMBERG
EJDE-2016/47
on Γ × (0, ∞), for every α ≥ 0, β ≥ 0, ω ∈ [0, 1), and where k : [0, ∞) → R
is a continuous nonnegative function, smooth on (0, ∞), vanishing at infinity and
satisfying the relation
Z
∞
k(s) ds = 1,
0
∂n represents the normal derivative and −∆Γ is the Laplace-Beltrami operator.
The cases ω = 0 and ω > 0 in (1.1) are usually referred as the Gurtin-Pipkin and
the Coleman-Gurtin models, respectively. The literature contains a full treatment
of equation (1.1) only in the case of standard boundary conditions (Dirichlet, Neumann and periodic boundary conditions). In light of new results and extensions for
the phase field equations (see, e.g., [2, 16] and references therein), we must consider
more general dynamic boundary conditions. In particular, we quote [22]:
In most works, the equations are endowed with Neumann boundary
conditions for both [unknowns] u and w (which means that the
interface is orthogonal to the boundary and that there is no mass
flux at the boundary) or with periodic boundary conditions. Now,
recently, physicists have introduced the so-called dynamic boundary
conditions, in the sense that the kinetics, i.e., ∂t u, appears explicitly
in the boundary conditions, in order to account for the interaction
of the components with the walls for a confined system.
The derivation of (1.2) in the context of (1.1) can be derived in a similar fashion
as in [15, 25] exploiting first and second laws of thermodynamics. Let ω ∈ [0, 1)
be fixed. It is clear that if we (formally) choose k = δ0 (the Dirac mass at zero),
equations (1.1)-(1.2) turn into the system
∂t u − ∆u + f (u) + α(1 − ω)u = 0,
in Ω × (0, ∞),
∂t u − ∆Γ u + ∂n u + g(u) + β(1 − ω)u = 0,
on Γ × (0, ∞).
(1.3)
(1.4)
The latter has been investigated quite extensively recently in many contexts (i.e.,
phase-field systems, heat conduction with a source at Γ, Stefan problems, etc).
Now we define, for ε ∈ (0, 1],
1 s
kε (s) = k( ),
ε ε
and we consider the same family of equations (1.1)-(1.2), replacing k with kε . Thus,
kε → δ0 when ε → 0. Our goal is to show in what sense does the system (1.1)-(1.2)
converge to (1.3)-(1.4) as ε → 0.
Such results seem to have begun with the hyperbolic relaxation of a ChaffeeInfante reaction diffusion equation in [28]. The motivation for such a hyperbolic
relaxation is similar to the motivation for applying a memory relaxation; it alleviates the parabolic problems from the sometimes unwanted property of “infinite
speed of propagation”. In [28] however, Hale and Raugel proved the existence of
a family of global attractors that is upper-semicontinuous in the phase space. A
global attractor is a unique compact invariant subset of the phase space that attracts all trajectories of the associated dynamical system, even at arbitrarily slow
rates (cf. [29] and [36, Theorem 14.6]). In a sense which will become clearer below,
upper-semicontinuity guarantees the attractors to not “blow-up” as the perturbation parameter vanishes; i.e.,
sup inf kx − ykXε → 0
x∈Aε y∈A0
as ε → 0+ .
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
3
Unlike global attractors, exponential attractors (sometimes called, inertial sets)
are compact positively invariant sets possessing finite fractal dimension that attract
bounded subsets of the phase space exponentially fast. It can readily be seen that
when both a global attractor A and an exponential attractor M exist, then A ⊆ M
provided that the basin of attraction of M is the whole phase space, and so the
global attractor is also finite dimensional. When we turn our attention to proving
the existence of exponential attractors, certain higher-order dissipative estimates
are required. In some interesting cases, it has not yet been shown how to obtain the
appropriate estimates (which would provide the existence of a compact absorbing
set, for example) independent of the perturbation parameter (cf. e.g. [11, 18]). It
is precisely because we are able to provide a higher-order uniform bound for the
model problems here that we do not give a separate upper-semicontinuity result
for the global attractors. An appropriate uniform higher-order bound will essentially/almost mean that a robustness result may be found (but it is not guaranteed).
Robust families of exponential attractors (that is, both upper- and lower-semicontinuous with explicit control over semidistances in terms of the perturbation parameter) of the type reported in [20] have successfully been shown to exist in many
different applications, of which we will limit ourselves to mention only [21] which
contains some applications of memory relaxation of reaction diffusion equations:
Cahn-Hilliard equations, phase-field equations, wave equations, beam equations,
and numerous others. The main idea behind robustness is typically an estimate of
the form
kSε (t)x − LS0 (t)ΠxkXε ≤ Cεp ,
(1.5)
for all t in some interval, where x ∈ Xε , Sε (t) : Xε → Xε and S0 (t) : X0 → X0
are semigroups generated by the solutions of the perturbed problem and the limit
problem, respectively, Π denotes a projection from Xε onto X0 and L is a “lift”
from X0 into Xε , and finally C, p > 0 are constants. Controlling this difference in a
suitable norm is crucial to obtaining our continuity results (see (C5) in Proposition
3.24). The estimate (1.5) means we can approximate the limit problem with the
perturbation with control explicitly written in terms of the perturbation parameter.
Usually such control is only exhibited on compact time intervals. Observe, a result
of this type will ensure that for every problem of type (1.3)-(1.4), there is an
“memory relaxation” of the form (1.1)-(1.2) close by in the sense that the difference
of corresponding trajectories satisfies (1.5).
We carefully treat the following issues:
(1) Well-posedness of the system comprising of equations (1.1)-(1.2) and (1.3)(1.4).
(2) Dissipation: the existence of bounded absorbing set, and a compact absorbing set, each of which is uniform with respect to the perturbation parameter
ε.
(3) Stability: existence of a family of exponential attractors for each ε ∈ [0, 1]
and an analysis of the continuity properties (robustness/Hölder) with respect to ε.
(4) The basin of attraction for each exponential attractor is the entire phase
space, and in demonstrating this result we see that the semigroup of solution
operators also admits a family of global attractors.
Concerning Issue 1, the well-posedness for a more general system, which includes
the one above, was given recently by [17]. The relevant results from that work are
4
J. L. SHOMBERG
EJDE-2016/47
cited below in Section 2. In this article we explore Issues 2, 3, and 4 in much more
depth; in particular, the existence of an exponential attractor for each ε ∈ [0, 1],
and the continuity of these attractors with respect to ε.
As is now customary (cf. [3, 6, 7, 27]) we introduce the so-called integrated past
history of u, i.e., the auxiliary variable
Z s
t
η (x, s) =
u(x, t − y) dy,
0
for s, t > 0. Setting
µ(s) = −(1 − ω)k 0 (s),
formal integration by parts into (1.1)-(1.2) yields
Z ∞
Z ∞
(1 − ω)
kε (s)∆u(x, t − s) ds =
µε (s)∆η t (x, s) ds,
0
0
Z ∞
Z ∞
(1 − ω)
kε (s)u(x, t − s) ds =
µε (s)η t (x, s) ds,
0
0
Z ∞
Z ∞
(1 − ω)
kε (s)∂n u(x, t − s) ds =
µε (s)∂n η t (x, s) ds,
0
0
Z ∞
Z ∞
(1 − ω)
kε (s)(−∆Γ + β)u(x, t − s) ds =
µε (s)(−∆Γ + β)η t (x, s) ds,
0
0
where
1 s
(1.6)
µ( ).
ε2 ε
For each ε ∈ (0, 1], the (perturbation) problem under consideration can now be
stated.
µε (s) =
Problem 1.1. Let α, β ≥ 0, and ω ∈ (0, 1). Find a function (u, η) such that
Z ∞
Z ∞
∂t u − ω∆u −
µε (s)∆η t (s) ds + α
µε (s)η t (s) ds + f (u) = 0
(1.7)
0
0
in Ω × (0, ∞), subject to the boundary conditions
Z ∞
∂t u − ω∆Γ u + ω∂n u +
µε (s)∂n η t (s) ds
0
Z ∞
+
µε (s)(−∆Γ + β)η t (s) ds + g(u) = 0
(1.8)
0
on Γ × (0, ∞), and
∂t η t (s) + ∂s η t (s) = u(t)
in Ω × (0, ∞),
(1.9)
with
η t (0) = 0
in Ω × (0, ∞),
(1.10)
and the initial conditions
u(0) = u0 in Ω, u(0) = v0 on Γ,
Z s
η 0 (s) = η0 :=
u0 (x, −y) dy in Ω, for s > 0,
Z 0s
η 0 (s) = ξ0 :=
v0 (x, −y) dy on Γ, for s > 0.
0
(1.11)
(1.12)
(1.13)
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
5
We will also discuss the problem corresponding to ε = 0. The results for this
problem may already be found in works in parabolic equations and the Wentzell
Laplacian (see [12, 13, 14, 19]). The singular (limit) problem is
Problem 1.2. Let α, β ≥ 0 and ω ∈ (0, 1). Find a function u such that
∂t u − ∆u + f (u) + α(1 − ω)u = 0
(1.14)
in Ω × (0, ∞), subject to the boundary conditions
∂t u − ∆Γ u + ∂n u + g(u) + β(1 − ω)u = 0
(1.15)
on Γ × (0, ∞), with the initial conditions
u(0) = u0
in Ω
and
u(0) = v0
on Γ.
(1.16)
Remark 1.3. It need not be the case that the boundary traces of u0 and η0 be equal
to v0 and ξ0 , respectively. Thus, we are solving a much more general problem in
which equation (1.7) is interpreted as an evolution equation in the bulk Ω properly
coupled with the equation (1.8) on the boundary Γ. Finally, from now on both η0
and ξ0 will be regarded as independent of the initial data u0 and v0 . Indeed, below
we will consider a more general problem with respect to the original one. This will
require a rigorous notion of solution to Problem (1.1) (cf. Definitions 2.1, 2.4),
hence we introduce the functional setting associated with this system.
Here below is the framework used to prove Hadamard well-posedness for Problem
(1.1). Consider the space X2 := L2 (Ω, dµ), where
dµ = dx|Ω ⊕ dσ,
where dx denotes the Lebesgue measure on Ω and dσ denotes the natural surface
measure on Γ. It is easy to see that X2 = L2 (Ω, dx) ⊕ L2 (Γ, dσ) may be identified
under the natural norm
Z
Z
2
2
kukX2 =
|u| dx + |u|2 dσ.
Ω
Γ
Moreover, if we identify every u ∈ C(Ω) with U = (u|Ω , u|Γ ) ∈ C(Ω) × C(Γ),
we may also define X2 to be the completion of C(Ω) in the norm k · kX2 . In
general, any function u ∈ X2 will be of the form u = uu12 with u1 ∈ L2 (Ω, dx)
and u2 ∈ L2 (Γ, dσ), and there need not be any connection between u1 and u2 .
From now on, the inner product in the Hilbert space X2 will be denoted by h·, ·iX2 .
Hereafter, the spaces L2 (Ω, dx) and L2 (Γ, dσ) will simply be denoted by L2 (Ω) and
L2 (Γ).
Recall that the Dirichlet trace map trD : C ∞ (Ω) → C ∞ (Γ), defined by trD (u) =
u|Γ extends to a linear continuous operator trD : H r (Ω) → H r−1/2 (Γ), for all
r > 1/2, which is onto for 1/2 < r < 3/2. This map also possesses a bounded
right inverse trD −1 : H r−1/2 (Γ) → H r (Ω) such that trD (trD −1 ψ) = ψ, for any
ψ ∈ H r−1/2 (Γ). We can thus introduce the subspaces of H r (Ω) × H r (Γ),
Vr := {(u, ψ) ∈ H r (Ω) × H r (Γ) : trD (u) = ψ},
(1.17)
for every r > 1/2, and note that we have the following dense and compact embeddings Vr1 ,→ Vr2 , for any r1 > r2 > 1/2. Finally, we think of V1 ' H 1 (Ω) ⊕ H 1 (Γ)
as the completion of C 1 (Ω) in the norm
Z
Z
2
2
2
kukV1 := (|∇u| + α|u| ) dx + (|∇Γ u|2 + β|u|2 )dσ
(1.18)
Ω
Γ
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J. L. SHOMBERG
EJDE-2016/47
(or some other equivalent norm in H 1 (Ω) × H 1 (Γ)). Naturally, the norm on the
space Vr is defined as
kuk2Vr := kuk2H r (Ω) + kuk2H r (Γ) .
(1.19)
For U = (u, u|Γ )tr ∈ V1 , let CΩ > 0 denote the best constant in which the
Sobolev-Poincaré inequality holds
ku − huiΓ kLs (Ω) ≤ CΩ k∇ukLs (Ω) ,
(1.20)
for s ≥ 1 (see [37, Lemma 3.1]). Here
1
huiΓ :=
|Γ|
Z
u|Γ dσ.
Γ
Let us now introduce the spaces for the memory variable η. For a nonnegative
measurable function θ defined on R+ and a real Hilbert space W (with inner product
denoted by h·, ·iW ), let L2θ (R+ ; W ) be the Hilbert space of W -valued functions on
R+ , endowed with the following inner product
Z ∞
hφ1 , φ2 iL2θ (R+ ;W ) :=
θ(s)hφ1 (s), φ2 (s)iW ds.
(1.21)
0
Consequently, for r > 1/2 we set
(
L2µε (R+ ; Vr ) for ε ∈ (0, 1],
r
Mε :=
{0}
when ε = 0,
and when r = 0 set
(
M0ε
:=
L2µε (R+ ; X2 ) for ε ∈ (0, 1],
{0}
when ε = 0.
One can see from [21, Lemma 5.1] that for ε1 ≥ ε2 > 0 and for fixed r = 0 or
r > 1/2, there holds the continuous embedding Mrε1 ,→ Mrε2 . As a matter of
convenience, the inner-product in M1ε is given by
Dη η E
1
2
,
ξ1
ξ2 M1ε
Z ∞
(1.22)
=
µε (s)(h∇η1 (s), ∇η2 (s)iL2 (Ω) + αhη1 (s), η(s)iL2 (Ω) ) ds
0
Z ∞
+
µε (s)(h∇Γ ξ1 (s), ∇Γ ξ2 (s)iL2 (Γ) + βhξ1 (s), ξ2 (s)iL2 (Γ) ) ds.
0
When it is convenient, we will use the notation
Hε0 := X2 × M1ε
(1.23)
Hε1
(1.24)
1
:= V ×
M2ε .
Each space is equipped with the corresponding “graph norm,” whose square is
defined by, for all ε ∈ [0, 1] and (U, Φ) ∈ Hεi , i = 0, 1,
k(U, Φ)k2H0ε := kU k2X2 + kΦk2M1ε
and k(U, Φ)k2H1ε := kU k2V1 + kΦk2M2ε .
For the kernel µ, we take the following assumptions (cf. e.g. [7, 23, 24]). Assume
µ ∈ C 1 (R+ ) ∩ L1 (R+ ),
µ(s) ≥ 0
for all
s ≥ 0,
(1.25)
(1.26)
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
µ0 (s) ≤ 0
0
µ (s) + δµ(s) ≤ 0
7
s ≥ 0,
(1.27)
for all s ≥ 0 and some δ > 0.
(1.28)
for all
The assumptions (1.25)-(1.27) are equivalent to assuming k(s) be a bounded, positive, nonincreasing, convex function of class C 2 . Moreover, assumption (1.28) guarantees exponential decay of the function µ(s) while allowing a singularity at s = 0.
Assumptions (1.25)-(1.27) are used in the literature (see [3, 7, 23, 27] for example)
to establish the existence and uniqueness of continuous global weak solutions to a
system of equations similar to (1.7), (1.9), but with Dirichlet boundary conditions.
In the literature, assumption (1.28) is used to obtain a bounded absorbing set for
the associated semigroup of solution operators.
For each ε ∈ (0, 1], define
D(Tε ) = {Φ ∈ M1ε : ∂s Φ ∈ M1ε , Φ(0) = 0}
(1.29)
where (with an abuse of notation) ∂s Φ is the distributional derivative of Φ and the
equality Φ(0) = 0 is meant in the following sense
lim kΦ(s)kX2 = 0.
s→0
Then define the linear (unbounded) operator Tε : D(Tε ) → M1ε by, for all Φ ∈
D(Tε ),
d
Tε Φ = − Φ.
ds
For each t ∈ [0, T ], the equation
∂t Φt = Tε Φt + U (t)
holds as an ODE in
M1ε
(1.30)
subject to the initial condition
Φ0 = Φ0 ∈ M1ε .
(1.31)
Concerning the solution to the IVP (1.30)-(1.31), we have the following proposition.
The result is a generalization of [27, Theorem 3.1].
Proposition 1.4. For each ε ∈ (0, 1], the operator Tε with domain D(Tε ) is an
infinitesimal generator of a strongly continuous semigroup of contractions on M1ε ,
denoted eTε t .
We now have (cf. e.g. [35, Corollary IV.2.2]).
Corollary 1.5. When U ∈ L1 ([0, T ]; V1 ) for each T > 0, then, for every Φ0 ∈ M1ε ,
the Cauchy problem
∂t Φt = Tε Φt + U (t), for t > 0,
(1.32)
Φ0 = Φ0 ,
has a unique solution Φ ∈ C([0, T ]; M1ε ) which can be explicitly given as (cf. [7,
Section 3.2] and [27, Section 3])
(R s
U (t − y) dy,
for 0 < s ≤ t,
t
0
Rt
Φ (s) =
(1.33)
Φ0 (s − t) + 0 U (t − y) dy, when s > t.
(The interested reader can also see [7, Section 3], [23, pp. 346–347] and [27, Section
3] for more details concerning the case with static boundary conditions.)
8
J. L. SHOMBERG
EJDE-2016/47
Furthermore, we also know that, for each ε ∈ (0, 1], Tε is the infinitesimal
generator of a strongly continuous (the right-translation) semigroup of contractions
on M1ε satisfying (1.34) below; in particular, Range(I − Tε ) = M1ε .
Following (1.28), there is the useful inequality. (Also see [7, see equation (3.4)]
and [27, Section 3, proof of Theorem].)
Corollary 1.6. There holds, for all Φ ∈ D(Tε ),
δ
hTε Φ, ΦiM1ε ≤ − kΦk2M1ε .
(1.34)
2ε
Even though the embedding V1 ,→ X2 is compact, it does not follow that the
embedding M1ε ,→ M0ε is also compact. Indeed, see [34] for a counterexample.
Moreover, this means the embedding Hε1 ,→ Hε0 is not compact. Such compactness
between the “natural phase spaces” is essential to the construction of finite dimensional exponential attractors. To alleviate this issue we follow [7, 21] and define for
any ε ∈ (0, 1] the so-called tail function of Φ ∈ M0ε by, for all τ ≥ 0,
Z
Tε (τ ; Φ) :=
εµε (s)kΦ(s)k2V1 ds,
(0,1/τ )∪(τ,∞)
With this we set, for ε ∈ (0, 1],
Kε2 := {Φ ∈ M2ε : ∂s Φ ∈ M0ε , Φ(0) = 0, sup τ Tε (τ ; Φ) < ∞}.
τ ≥1
The space
Kε2
is Banach with the norm whose square is defined by
kΦk2Kε2 := kΦk2M2ε + εk∂s Φk2M0ε + sup τ Tε (τ ; Φ).
(1.35)
τ ≥1
When ε = 0, we set K02 = {0}. Importantly, for each ε ∈ (0, 1], the embedding
Kε2 ,→ M1ε is compact. (cf. [21, Proposition 5.4]). Hence, let us now also define the
space
Vε1 := V1 × Kε2 ,
and the desired compact embedding Vε1 ,→ Hε0 holds. Again, each space is equipped
with the corresponding graph norm whose square is defined by, for all ε ∈ [0, 1] and
(U, Φ) ∈ Vε1 ,
k(U, Φ)k2Vε1 := kU k2V1 + kΦk2Kε2 .
In regards to the system in Corollary 1.5 above, we will also call upon the
following simple generalizations of [7, Lemmas 3.3, 3.4, and 3.6].
Lemma 1.7. Let ε ∈ (0, 1] and Φ0 ∈ D(Tε ). Assume there is ρ > 0 such that, for
all t ≥ 0, kU (t)kV1 ≤ ρ. Then for all t ≥ 0,
εkTε Φt k2M1ε ≤ εe−δt kTε Φ0 k2M1ε + ρ2 kµkL1 (R+ ) .
(1.36)
Remark 1.8. The above result will also be needed later in the weaker space M0ε
(see Step 3 in the proof of Lemma 3.13). The result for the weaker space can be
obtained by suitably transforming (1.32)-(1.33) and applying an appropriate bound
on U .
Lemma 1.9. Let ε ∈ (0, 1] and Φ0 ∈ D(Tε ). Assume there is ρ > 0 such that, for
all t ≥ 0, kU (t)kV1 ≤ ρ. Then there is a constant C > 0 such that, for all t ≥ 0,
sup τ Tε (τ ; Φt ) ≤ 2(t + 2)e−δt sup τ Tε (τ ; Φ0 ) + Cρ2 .
τ ≥1
τ ≥1
(1.37)
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BOUNDARY CONDITIONS WITH MEMORY
9
Finally, we give a version of Lemma 1.9 for compact intervals.
Lemma 1.10. Let ε ∈ (0, 1], T > 0, and Φ0 ∈ D(Tε ). Assume there is ρ > 0 such
that
Z T
kU (τ )k2V1 dτ ≤ ρ.
0
Then there is a positive constant C(T ) such that, for all t ∈ [0, T ],
sup τ Tε (τ ; Φt ) ≤ C(T ) ρ + sup τ Tε (τ ; Φ0 ) .
τ ≥1
τ ≥1
We now discuss the linear operator associated with the model problem. In our
case it is given by the following (note that in [7, Section 3.1] the basic tool is the
Laplacian with Dirichlet boundary conditions; in our case, the analogue operator
turns out to be the so-called “Wentzell” Laplace operator).
Proposition 1.11. Let Ω be a bounded open set of Rn with Lipschitz boundary Γ.
2
For α, β ≥ 0, define the operator Aα,β
W on X , by
−∆ + αI
0
α,β
AW :=
,
(1.38)
∂n (·)
−∆Γ + βI
with
n
tr
1
2
D(Aα,β
W ) := U = (u1 , u2 ) ∈ V : −∆u1 + αu1 ∈ L (Ω),
o
∂n u1 − ∆Γ u2 + βu2 ∈ L2 (Γ) .
(1.39)
α,β
2
Then, (Aα,β
W , D(AW )) is self-adjoint and nonnegative operator on X whenever
α,β
α, β ≥ 0, and AW > 0 (is strictly positive) if either α > 0 or β > 0. Moreover, the
−1
resolvent operator (I+Aα,β
∈ L(X2 ) is compact. If the boundary Γ is of class C 2 ,
W )
α,β
2
then D(AW ) = V (see, e.g., [2, Theorem 2.3]). Indeed, for any α, β ≥ 0, the map
2
2
2
2
Ψ : U 7→ Aα,β
W U , when viewed as a map from V into X = L (Ω) × L (Γ), is an
isomorphism, and there exists a positive constant C∗ , independent of U = (u, ψ)tr ,
such that
C∗−1 kU kV2 ≤ kΨ(U )kX2 ≤ C∗ kU kV2 ,
(1.40)
2
for all U ∈ V (cf. Lemma 4.1).
We can refer the reader to [4] for an extensive survey of recent results concerning
the “Wentzell” Laplacian Aα,β
W .
For the nonlinear terms, assume f, g ∈ C 1 (R) satisfy the growth assumptions:
there exist positive constants `1 and `2 , and r1 , r2 ∈ [1, 25 ) such that for all s ∈ R,
|f 0 (s)| ≤ `1 (1 + |s|r1 ),
0
r2
|g (s)| ≤ `2 (1 + |s| ).
(1.41)
(1.42)
We also assume there are positive constants Mf and Mg so that for all s ∈ R,
f 0 (s) > −Mf ,
0
g (s) > −Mg .
(1.43)
(1.44)
Consequently, (1.43)-(1.44) imply there are κi > 0, i = 1, 2, 3, 4, so that for all
s ∈ R,
f (s)s ≥ −κ1 s2 − κ2 ,
(1.45)
10
J. L. SHOMBERG
EJDE-2016/47
g(s)s ≥ −κ3 s2 − κ4 .
(1.46)
Remark 1.12. Observe that here we do not allow for the critical polynomial growth
exponent (of 5) which appears in several works with static boundary conditions (cf.
e.g. [3, 7]). Indeed, in order for us to obtain a notion of strong solution (see
Definition 2.4 below), the arguments in the proof of Theorem 2.6 do not allow for
ri ≥ 5/2, i = 1, 2.
We can follow [7, Section 4] or, more precisely [23, 24] to deduce the existence
and uniqueness of weak solutions in the above class exploiting both semigroup
methods and energy methods in the framework of a Galerkin scheme which can
be constructed for problems with dynamic boundary conditions (see, [2, Theorem
2.3]).
Constants appearing below are independent of ε and ω, unless specified otherwise, but may depend on various structural parameters such as α, β, |Ω|, |Γ|, `f
and `g , and the constants may even change from line to line. We denote by Q(·) a
generic monotonically increasing function. We will use kBkW := supΥ∈B kΥkW to
denote the “size” of the subset B in the Banach space W .
2. Review of well-posedness and regularity
Here we provide some definitions and cite the relevant global well-posedness
results concerning Problem (1.1). For the remainder of this article we choose to set
n = 3, which is of course the most relevant physical dimension.
Below we will set F : R2 → R2 ,
f (u)
F (U ) :=
,
(2.1)
ge(u)
where ge(s) := g(s)−ωβs, for s ∈ R. (To offset ge, the term ωβu will be incorporated
0,β
in the operator A0,0
W as AW .)
Definition 2.1. Let ε ∈ (0, 1], ω ∈ (0, 1) and T > 0. Given U0 = (u0 , v0 )tr ∈ X2
and Φ0 = (η0 , ξ0 )tr ∈ M1ε , the pair U (t) = (u(t), v(t))tr and Φt = (η t , ξ t )tr satisfying
U ∈ L∞ ([0, T ]; X2 ) ∩ L2 ([0, T ]; V1 ),
(2.2)
r1
u ∈ L (Ω × [0, T ]),
(2.3)
r2
v ∈ L (Γ × [0, T ]),
∞
Φ∈L
1 ∗
2
(2.4)
([0, T ]; M1ε ),
r10
(2.5)
r20
∂t U ∈ L ([0, T ]; (V ) ) ⊕ (L (Ω × [0, T ]) × L (Γ × [0, T ])),
(2.6)
∂t Φ ∈ L2 ([0, T ]; Hµ−1
(R+ ; V1 )),
ε
(2.7)
t
is said to be a weak solution to Problem (1.1) if, v(t) = u|Γ (t) and ξ = η t |Γ
for almost all t ∈ [0, T ], and for all Ξ = (ς, ς|Γ )tr ∈ V1 ∩ (Lr1 (Ω) × Lr2 (Γ)),
Π = (ρ, ρ|Γ )tr ∈ M1ε , and for almost all t ∈ [0, T ], there holds,
t
h∂t U (t), ΞiX2 + ωhA0,β
W U (t), ΞiX2 + hΦ , ΞiM1ε + hF (U (t)), ΞiX2 = 0,
t
t
h∂t Φ , ΠiM1ε = hTε Φ , ΠiM1ε + hU (t), ΠiM1ε ,
(2.8)
(2.9)
in addition,
U (0) = U0
and
Φ0 = Φ0 .
(2.10)
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BOUNDARY CONDITIONS WITH MEMORY
11
The function [0, T ] 3 t 7→ (U (t), Φt ) is called a global weak solution if it is a weak
solution for every T > 0.
Remark 2.2. When we have a weak solution to Problem (1.1), the above restrictions u|Γ (t) and η|tΓ are well-defined by virtue of the Dirichlet trace map,
trD : H 1 (Ω) → H 1/2 (Γ). However, this is not necessarily the case for ∂t U .
Remark 2.3. The continuity properties U ∈ C([0, T ]; X2 ) follow from the classical
embedding (cf. e.g. [38, Lemma 5.51]),
{χ ∈ L2 ([0, T ]; V ), ∂t χ ∈ L2 ([0, T ]; V 0 )} ,→ C([0, T ]; H),
where H and V are reflexive Banach spaces with continuous embeddings V ,→
H ,→ V 0 , the injection V ,→ H being compact.
Definition 2.4. The pair U (t) = (u(t), v(t))tr and Φt = (η t , ξ t )tr is called a (global)
strong solution of Problem (1.1) if it is a weak solution in the sense of Definition
2.1, and if it satisfies the following regularity properties:
U ∈ L∞ ([0, ∞); V1 ) ∩ L2 ([0, ∞); V2 ),
∞
Φ∈L
∞
([0, ∞); M2ε ),
2
2
(2.12)
1
∂t U ∈ L ([0, ∞); X ) ∩ L ([0, ∞); V ),
∂t Φ ∈ L
∞
(2.11)
([0, ∞); M1ε ).
(2.13)
(2.14)
Therefore, (U (t), Φt ) satisfies the equations (2.8)-(2.9) almost everywhere, i.e., is a
strong solution.
Theorem 2.5 (Weak solutions). Assume (1.25)-(1.27) and (1.41)-(1.44) hold. For
each ε ∈ (0, 1], ω ∈ (0, 1) and T > 0, and for any U0 = (u0 , v0 )tr ∈ X2 and
Φ0 = (η0 , ξ0 )tr ∈ M1ε , there exists a unique (global) weak solution to Problem (1.1)
in the sense of Definition 2.1 which depends continuously on the initial data in the
following way; there exists a constant C > 0, independent of Ui , Φi , i = 1, 2, and
T > 0 in which, for all t ∈ [0, T ], there holds
kU1 (t) − U2 (t)kX2 + kΦt1 − Φt2 kM1ε ≤ (kU1 (0) − U2 (0)kX2 + kΦ01 − Φ02 kM1ε )eCt . (2.15)
Proof. Cf. [17, Theorem 3.8] for existence and [17, Proposition 3.10] for (2.15). We conclude the preliminary results for Problem (1.1) with the following result.
Theorem 2.6 (Strong solutions). Assume (1.25)–(1.27) and (1.41)–(1.44) hold.
For each ε ∈ (0, 1], ω ∈ (0, 1), and T > 0, and for any U0 = (u0 , v0 )tr ∈ V1 and
Φ0 = (η0 , ξ0 )tr ∈ M2ε , there exists a unique (global) strong solution to Problem (1.1)
in the sense of Definition 2.4.
For a proof of the above theorem see [17, Theorem 3.11]. Here we recall some
important aspects and relevant results for Problem (1.2). The interested reader can
also see [12, 13, 14, 19] for further details.
Definition 2.7. Let ω ∈ (0, 1) and T > 0. Given U0 = (u0 , v0 )tr ∈ X2 , the pair
U (t) = (u(t), v(t))tr satisfying
U ∈ L∞ ([0, T ]; X2 ) ∩ L2 ([0, T ]; V1 ),
(2.16)
r1
(2.17)
r2
(2.18)
u ∈ L (Ω × [0, T ]),
v ∈ L (Γ × [0, T ]),
12
J. L. SHOMBERG
0
EJDE-2016/47
0
∂t U ∈ L2 ([0, T ]; (V1 )∗ ) ⊕ (Lr1 (Ω × [0, T ]) × Lr2 (Γ × [0, T ])),
(2.19)
is said to be a weak solution to Problem (1.2) if, v(t) = u|Γ (t) for almost all
t ∈ [0, T ], and for all Ξ = (ς, ς|Γ )tr ∈ V1 ∩ (Lr1 (Ω) × Lr2 (Γ)), and for almost all
t ∈ [0, T ], there holds
h∂t U (t), ΞiX2 + ωhA0,β
W U (t), ΞiX2 + hF (U (t)), ΞiX2 = 0,
(2.20)
with
U (0) = U0 .
(2.21)
The function [0, T ] 3 t 7→ U (t) is called a global weak solution if it is a weak solution
for every T > 0.
We remind the reader of Remark 2.2 on the issue of traces. We conclude this
section with the following result.
Theorem 2.8 (Weak solutions). Assume (1.41)-(1.44) hold. For each ω ∈ (0, 1)
and T > 0, and for any U0 = (u0 , v0 )tr ∈ X2 , there exists a unique (global) weak
solution to Problem (1.2) in the sense of Definition 2.7 which depends continuously
on the initial data as follows: there exists a constant C > 0, independent of U1 and
U2 , and T > 0 in which, for all t ∈ [0, T ], there holds
kU1 (t) − U2 (t)kX2 ≤ kU1 (0) − U2 (0)kX2 eCt .
(2.22)
For a proof of the above theorem see [13, Theorem 2.2].
3. Asymptotic behavior and attractors
3.1. Preliminary estimates. Concerning Problem (1.1) and following directly
from Theorem 2.5, we have the first preliminary result for this section.
Corollary 3.1. Problem (1.1) defines a (nonlinear) strongly continuous semigroup
Sε (t) on the phase space Hε0 = X2 × M1ε by
Sε (t)Υ0 := (U (t), Φt ),
where Υ0 = (U0 , Φ0 ) ∈ Hε0 and (U (t), Φt ) is the unique solution to Problem (1.1).
The semigroup is Lipschitz continuous on Hε0 via the continuous dependence estimate (2.15).
The next preliminary result concerns a uniform bound on the weak solutions.
This result follows from an estimate which proves the existence of a bounded absorbing set for the semigroup of solution operators. This result provides a basic but
important first step in showing the associated dynamical system is dissipative (cf.
e.g. [1, 39]). It is important to note that throughout the remainder of this article,
whereby we are now concerned with the asymptotic behavior of the solutions to
Problem (1.1) and Problem (1.2),
(A1) we will assume that (1.28) holds.
Additionally, we introduce a smallness criteria for certain parameters relating to
the linear operator Aα,β
W and the nonlinear map F .
(A2) Smallness criteria: Fix ε ∈ (0, 1] and ω ∈ (0, 1). Denote by CΩ the positive
constant that arises from the embedding V1 ,→ X2 ; i.e., kU k2X2 ≤ CΩ kU k2V1 .
The smallness criteria is that κ1 , κ3 , β > 0 (cf. (1.38) and (1.45)-(1.46))
satisfy
max{κ1 , κ3 + β} < ωCΩ−1 .
(3.1)
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BOUNDARY CONDITIONS WITH MEMORY
13
As a final note, we remind the reader that all formal multiplication below can be
rigorously justified using the Galerkin procedure developed in the proof of Theorem
2.5 in [17].
Lemma 3.2. Let ε ∈ (0, 1] and ω ∈ (0, 1). In addition to the assumptions of Theorem 2.5, assume (1.28) holds and that κ1 , κ3 , β > 0 satisfy the smallness criteria
(3.1). For all R > 0 and Υ0 = (U0 , Φ0 ) ∈ Hε0 = X2 × M1ε with kΥ0 kH0ε ≤ R
for all ε ∈ (0, 1], there exist positive constants ν0 = ν0 (ω, CΩ , κ1 , κ3 , β, δ) and
P0 = P0 (κ2 , κ4 , ν0 ), and there is a positive monotonically increasing function Q(·)
each independent of ε, in which, for all t ≥ 0,
k(U (t), Φt )k2H0ε ≤ Q(R)e−ν0 t + P0 .
(3.2)
p
Bε0 := (U, Φ) ∈ Hε0 : k(U, Φ)kH0ε ≤ P0 + 1 .
(3.3)
Moreover, the set
is absorbing and positively invariant for the semigroup Sε (t).
Proof. Let ε ∈ (0, 1] and ω ∈ (0, 1). Let Υ0 = (U0 , Φ0 ) ∈ Hε0 = X2 × M1ε . From the
equations (2.8) and (2.9), we take the corresponding weak solution Ξ = U (t) and
Π(s) = Φt (s). We then obtain the identities
t
h∂t U, U iX2 + ωhA0,β
W U, U iX2 + hΦ , U iM1ε + hF (U ), U iX2 = 0,
t
t
t
t
t
h∂t Φ , Φ iM1ε = hTε Φ , Φ iM1ε + hU, Φ iM1ε .
(3.4)
(3.5)
Observe that
1 d
kU k2X2 ,
2 dt
= k∇uk2L2 (Ω) + k∇Γ uk2L2 (Γ) + βkuk2L2 (Γ) ,
h∂t U, U iX2 =
hA0,β
W U, U iX2
(3.6)
(3.7)
1 d
kΦt k2M1ε .
(3.8)
2 dt
Combining (3.4)-(3.8) produces the differential identity, which holds for almost all
t ≥ 0,
1 d
kU k2X2 + kΦt k2M1ε
2 dt
(3.9)
+ ω(k∇uk2L2 (Ω) + k∇Γ uk2L2 (Γ) + βkuk2L2 (Γ) )
h∂t Φt , Φt iM1ε =
− hTε Φt , Φt iM1ε + hF (U ), U iX2 = 0.
Because of assumption (1.28), we may directly apply (1.34) from Corollary 1.6; i.e.,
− hTε Φt , Φt iM1ε ≥
δ
kΦt k2M1ε .
2ε
(3.10)
From (1.45) and (1.46), we know that
hF (U ), U iX2 ≥ −κ1 kuk2L2 (Ω) − (κ3 + ωβ)kuk2L2 (Γ) − (κ2 + κ4 )
≥ −κ1 kuk2L2 (Ω) − (κ3 + β)kuk2L2 (Γ) − (κ2 + κ4 )
(3.11)
= −CF kU k2X2 − (κ2 + κ4 ),
where CF := max{κ1 , κ3 + β}. Finally, due the embedding V1 ,→ X2 , we have
CΩ−1 kU k2X2 ≤ kU k2V1 ,
(3.12)
14
J. L. SHOMBERG
EJDE-2016/47
for some CΩ > 0. Hence, (3.9)-(3.12) yields the differential inequality (minimizing
the left-hand side by setting ε = 1),
d
kU k2X2 + kΦt k2M1ε + 2(ωCΩ−1 − CF )kU k2X2 + δkΦt k2M1ε ≤ 2(κ2 + κ4 ).
dt
By the smallness criteria (3.1) there holds
ωCΩ−1 − CF > 0.
Thus we arrive at the differential inequality, which holds for almost all t ≥ 0,
d
(3.13)
kU k2X2 + kΦt k2M1ε + m0 (kU k2X2 + kΦt k2M1ε ) ≤ C.
dt
where m0 := min{2(ωCΩ−1 − CF ), δ} > 0, and C > 0 depends only on κ2 and κ4 .
(The absolute continuity of the mapping t 7→ kU (t)k2X2 + kΦt k2M1 can be estabε
lished as in [39, Lemma III.1.1], for example.) After applying a suitable Grönwall
inequality, the estimate (3.2) follows with ν0 = m0 and P0 = mC0 ; indeed, (3.13)
yields, for all t ≥ 0,
kU (t)k2X2 + kΦt k2M1ε ≤ e−m0 t kU0 k2X2 + kΦ0 k2M1ε + P0 .
(3.14)
Now we see (3.2) holds for any R > 0 and Υ0 = (U0 , Φ0 ) ∈ Hε0 such that kΥ0 kH0ε ≤
R for all ε ∈ (0, 1].
The existence of the bounded set Bε0 in Hε0 that is absorbing and positively
invariant for Sε (t) follows from (3.14) (cf. e.g. [31, Proposition 2.64]). Given any
nonempty bounded subset B in Hε0 \ Bε0 , then we have that Sε (t)B ⊆ B 0 , in Hε0 ,
for all t ≥ t0 where
1
t0 ≥
ln kBk2H0ε .
(3.15)
m0
(Observe that t0 > 0 because kBkH0ε > 1.) This completes the proof.
Corollary 3.3. From (3.2) it follows that for each ε ∈ (0, 1] and ω ∈ (0, 1), any
weak solution (U (t), Φt ) to Problem (1.1), according to Definition 2.1, is bounded
uniformly in t. Indeed, for all Υ0 ∈ Hε0 ,
lim sup kSε (t)Υ0 kH0ε ≤ Pe0 ,
(3.16)
t→+∞
where Pe0 depends on P0 and the initial datum.
Corollary 3.4. Problem (1.1) defines a (nonlinear) strongly continuous semigroup
Sε (t) on the phase space Hε0 = X2 × M1ε by
Sε (t)Υ0 := (U (t), Φt ),
where Υ0 = (U0 , Φ0 ) ∈ Hε0 and (U (t), Φt ) is the unique solution to Problem (1.1).
The semigroup is Lipschitz continuous on Hε0 via the continuous dependence estimate (2.15).
Remark 3.5. Thanks to the uniformity of the above estimates with respect to the
perturbation parameter ε, it is easy to see that there exists a bounded absorbing
set B00 for the semigroup S0 : H00 = X2 → X2 generated by the weak solutions of
Problem (1.2). Moreover, we also easily see that Problem (1.2) defines a semigroup
S0 (t) : H00 = X2 → X2 by S0 (t)U0 := U (t). (See the references mentioned above for
further details.)
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BOUNDARY CONDITIONS WITH MEMORY
15
3.2. Exponential attractors. Exponential attractors (sometimes called inertial
sets) are positively invariant sets possessing finite fractal dimension that attract
bounded subsets of their basin of attraction exponentially fast. This section will
focus on the existence of exponential attractors. The existence of an exponential
attractor depends on certain properties of the semigroup; namely, the smoothing
property for the difference of any two trajectories and the existence of a more regular
bounded absorbing set in the phase space (see e.g. [8, 9, 20] and in particular [7, 21]).
The basin of attraction will be discussed in the next section.
The main result of this section is the following.
Theorem 3.6. For each ε ∈ [0, 1] and ω ∈ (0, 1), the dynamical system (Sε , Hε0 )
associated with Problem (1.1) admits an exponential attractor Mε compact in Hε0 ,
and bounded in Vε1 . Moreover,
(i) For each t ≥ 0, Sε (t)Mε ⊆ Mε .
(ii) The fractal dimension of Mε with respect to the metric Hε0 is finite, uniformly in ε; namely,
dimF (Mε , Hε0 ) ≤ C < ∞,
for some positive constant C independent of ε.
(iii) There exist % > 0 and a positive nondecreasing function Q such that, for
all t ≥ 0,
distH0ε (Sε (t)B, Mε ) ≤ Q(kBkH0ε )e−%t ,
for every nonempty bounded subset B of Hε0 .
Remark 3.7. Above, the fractal dimension of Mε in Hε0 is given by
dimF (Mε , Hε0 ) := lim sup
r→0
ln µH0ε (Mε , r)
<∞
− ln r
where µH0ε (X , r) denotes the minimum number of r-balls from Hε0 required to cover
X.
The proof of Theorem 3.6 follows from the application of an abstract result
reported here for our problem (see e.g. [7, 21]; cf. also Remark 3.16 below).
Proposition 3.8. Let (Sε , Hε0 ) be a dynamical system for each ε ∈ [0, 1]. Assume
the following hypotheses hold:
(C1) There exists a bounded absorbing set Bε1 ⊂ Vε1 which is positively invariant
for Sε (t). More precisely, there exists a time t1 > 0, uniform in ε, such
that
Sε (t)Bε1 ⊂ Bε1
for all t ≥ t1 where Bε1 is endowed with the topology of Hε0 .
(C2) There is t∗ ≥ t1 such that the map Sε (t∗ ) admits the decomposition, for
each ε ∈ (0, 1] and for all Υ0 , Ξ0 ∈ Bε1 ,
Sε (t∗ )Υ0 − Sε (t∗ )Ξ0 = Lε (Υ0 , Ξ0 ) + Rε (Υ0 , Ξ0 )
where, for some constants α∗ ∈ (0, 21 ) and Λ∗ = Λ∗ (Ω, t∗ , ω) ≥ 0, the
following hold:
kLε (Υ0 , Ξ0 )kH0ε ≤ α∗ kΥ0 − Ξ0 kH0ε ,
∗
kRε (Υ0 , Ξ0 )kVε1 ≤ Λ kΥ0 − Ξ0 kH0ε .
(3.17)
(3.18)
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J. L. SHOMBERG
EJDE-2016/47
(C3) The map
(t, Υ) 7→ Sε (t)Υ : [t∗ , 2t∗ ] × Bε1 → Bε1
is Lipschitz continuous on Bε1 in the topology of Hε0 .
Then (Sε , Hε0 ) possesses an exponential attractor Mε in Bε1 .
We now prove the hypotheses of Proposition 3.8 and we again remind the reader
that for the remainder of the article, we assume that the smallness criteria (3.1)
holds, in addition to the assumption (1.28). We begin with the perturbation Problem (1.1). The results for the singular Problem (1.2) will follow.
Lemma 3.9. Condition (C1) holds for each ε ∈ (0, 1] and ω ∈ (0, 1). Moreover,
for all R > 0 and Υ0 = (U0 , Φ0 ) ∈ Vε1 = V1 × Kε2 with kΥ0 kVε1 ≤ R for all ε ∈
(0, 1], there exists a positive constant P1 = P1 (ν1 , Pe0 ) and a positive monotonically
increasing function Q(·), each independent of ε, such that, for all t ≥ 0,
k(U (t), Φt )k2Vε1 ≤ Q(R)e− min{δ,1}t (t + 1) + 2P1 .
(3.19)
Proof. Let ε ∈ (0, 1], ω ∈ (0, 1) and Υ0 = (U0 , Φ0 ) ∈ Vε1 = V1 × Kε2 . For all s, t ≥ 0,
α,β t
t
let Z(t) = Aα,β
W U (t) and Θ (s) = AW Φ (s). In equations (2.8)-(2.9), take Ξ = Z(t)
t
and Π = Θ (s). Proceeding as in [17, proof of Theorem 3.11] (however, this time
we are able to enjoy the uniform bounds (2.11)), we obtain the identities
t
h∂t U, ZiX2 + ωhA0,β
W U, ZiX2 + hΦ , ZiM1ε + hF (U ), ZiX2 = 0,
t
t
t
t
t
h∂t Φ , Θ iM1ε = hTε Φ , Θ iM1ε + hU, Θ iM1ε .
(3.20)
(3.21)
These two identities may be combined together after we observe that, from the
definition of the product given in (1.22),
Z ∞
t
hΦ , ZiM1ε =
µε (s)hΦt (s), ZiV1 ds
0
Z ∞
t
=
µε (s)hAα,β
W Φ (s), ZiX2 ds
0
Z ∞
α,β
t
=
µε (s)hAα,β
W Φ (s), AW U iX2 ds
(3.22)
Z0 ∞
=
µε (s)hΘt (s), Aα,β
W U iX2 ds
0
Z ∞
=
µε (s)hΘt (s), U iV1 ds
0
= hU, Θt iM1ε .
Now inserting (3.22) into (3.20) and adding the result to (3.21), we obtain the
identity
t
t
h∂t U, ZiX2 + ωhA0,β
W U, ZiX2 + h∂t Φ , Θ iM1ε
− hTε Φt , Θt iM1ε + hF (U ), ZiX2 = 0.
(3.23)
Next we write
h∂t U, ZiX2 = h∂t U, Aα,β
W U iX2 = h∂t U, U iV1 =
1 d
kU k2V1 ,
2 dt
(3.24)
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BOUNDARY CONDITIONS WITH MEMORY
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ωhA0,β
W U, ZiX2 = ω(h∇u, ∇ziL2 (Ω) + h∇Γ u, ∇Γ ziL2 (Γ) + βhv, ziL2 (Γ) )
= ωhAα,β
W U, ZiX2 − ωαhu, ziL2 (Ω)
=
ωkZk2X2
(3.25)
− ωαhu, ziL2 (Ω) ,
and
t
Z
t
h∂t Φ , Θ iM1ε =
∞
µε (s)h∂t Φt (s), Θt (s)iV1 ds
0
Z
∞
=
Z0 ∞
=
t
t
µε (s)h∂t Aα,β
W Φ (s), Θ (s)iX2 ds
µε (s)h∂t Θt (s), Θt (s)iX2 ds
(3.26)
0
= h∂t Θt , Θt iM0ε
=
1 d
kΘt k2M0ε .
2 dt
Combining (3.23)-(3.26) brings us to the differential identity, which holds for almost
all t ≥ 0,
1 d
kU k2V1 + kΘt k2M0ε + ωkZk2X2 − hTε Φt , Θt iM1ε + hF (U ), ZiX2
2 dt
= ωαhu, ziL2 (Ω) .
(3.27)
With assumption (1.28) we are able to obtain
hTε Φt , Θt iM1ε =
∞
Z
µε (s)hTε Φt (s), Θt (s)iV1 ds
0
∞
Z
t
t
µε (s)hTε Aα,β
W Φ (s), Θ (s)iX2 ds
=
0
∞
Z
µε (s)hTε Θt (s), Θt (s)iX2 ds
=
0
Z
∞
d t
Θ (s), Θt (s)iX2 ds
ds
0
Z
1 ∞
d
=−
µε (s) kΘt (s)k2X2 ds
2 0
ds
Z
Z
1 ∞ 0
1 ∞ d
(µε (s)kΘt (s)k2X2 ) ds +
µε (s)kΘt (s)k2X2 ds
=−
2 0 ds
2 0
{z
}
|
=0
Z
δ ∞
≤−
µε (s)kΘt (s)k2X2 ds
2 0
δ
= − kΘt k2M0ε .
2
(3.28)
=−
µε (s)h
18
J. L. SHOMBERG
EJDE-2016/47
Multiplying the nonlinear term by Z in X2 produces, with an application of integration by parts,
hF (U ), ZiX2
Z
Z
=
f (u)(−∆u + αu) dx + ge(u)(−∆Γ u + ∂n u + βu)dσ
Γ
ZΩ
Z
0
2
0
=
f (u)|∇u| dx + ge (u)|∇Γ u|2 dσ
Ω
Z
ZΓ
Z
+α
f (u)u dx + β ge(u)udσ + (e
g (u) − f (u))∂n udσ.
Ω
Γ
(3.29)
Γ
Directly from (1.43) and (1.44), we see that
Z
Z
f 0 (u)|∇u|2 dx + ge0 (u)|∇Γ u|2 dσ ≥ −Mf k∇uk2L2 (Ω) − Mg k∇Γ uk2L2 (Γ) , (3.30)
Γ
Ω
and from (1.45)-(1.46), we obtain
Z
Z
α
f (u)u dx + β ge(u)udσ
Ω
Γ
Z
Z
Z
=α
f (u)u ds + β g(u)udσ − ωβ 2 u2 dσ
Ω
≥
≥
Γ
−ακ1 kuk2L2 (Ω) −
−C(kU k2X2 + 1),
ακ2 −
Γ
βκ3 kuk2L2 (Γ)
− βκ4 − ωβ
(3.31)
2
kuk2L2 (Γ)
for some constant C > 0, independent of t. For the last term in (3.29) we recall [17,
Proof of Theorem 3.11]. Due to the assumptions (1.41)-(1.42) it suffices to bound
boundary integrals of the form, for some r < 52 ,
Z
I :=
ur+1 ∂n udσ.
Γ
Indeed, thanks to the trace and regularity embeddings, for all ω ∈ (0, 1) and for
some Cω ∼ C
ω > 0,
ω
I ≤ k∂n ukH 1/2 (Γ) kur+1 kH −1/2 (Γ) ≤ kuk2H 2 (Ω) + Cω kur+1 k2H −1/2 (Γ) .
(3.32)
4
To bound the last term in (3.32) we will employ the Sobolev embeddings (recall
Γ is two-dimensional) H 1/2 (Γ) ,→ L4 (Γ) and H 1 (Γ) ,→ Ls (Γ), for any s ∈ ( 43 , ∞).
Then, by employing some basic Hölder inequalities
kur+1 k2H −1/2 (Γ) =
≤
sup
|hur+1 , ψiL2 (Γ) |2
(3.33)
kur+1 ψk2L1 (Γ)
(3.34)
ψ∈H 1/2 (Γ):kψkH 1/2 (Γ) =1
sup
ψ∈H 1/2 (Γ):kψkH 1/2 (Γ) =1
≤ kuk2Ls (Γ) kuk2r
Lsr (Γ)
(3.35)
≤ Ckuk2H 1 (Γ) kuk2r
Lsr (Γ) ,
(3.36)
for some positive constant C and for sufficiently large s ∈ ( 43 , ∞), where s :=
4s/(3s − 4) > 4/3. Next we exploit the interpolation inequality
1/2r
1−1/2r
kukLsr (Γ) ≤ CkukH 2 (Γ) kukL2 (Γ) ,
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
19
provided that r = 1 + 2/s < 5/2, where we further infer from (3.36) that
2r−1
kur+1 k2H −1/2 (Γ) ≤ Ckuk2H 1 (Γ) kukH 2 (Γ) kukL
2 (Γ)
2(2r−1)
2
2
≤ ηkukH 2 (Γ) + Cη kukH 1 (Γ) kuk2H 1 (Γ) kukL2 (Γ) ,
(3.37)
for any η ∈ (0, 1]. Inserting (3.37) into (3.32) and choosing a sufficiently small
η = ω/Cω , by virtue of (1.40), we easily deduce
ω
2(2r−1)
I ≤ kZk2X2 + Cω kuk2H 1 (Γ) kuk2H 1 (Γ) kukL2 (Γ) .
(3.38)
4
Together, (3.30)-(3.38) provide the following bound on (3.29) for all ω > 0, and for
some positive constants C and Cω ∼ C
ω,
ω
hF (U ), ZiX2 ≥ −C(kU k2X2 + 1) − kZk2X2
4
(3.39)
2(2r−1)
− Cω kuk2H 1 (Γ) kuk2H 1 (Γ) kukL2 (Γ)
.
Also with Young’s inequality,
ω
kzk2L2 (Ω)
4
(3.40)
ω
≤ ωα2 kuk2L2 (Ω) + kZk2X2 .
4
Applying the estimates (3.28), (3.39) and (3.40) to (3.27), we arrive at the differential inequality, which holds for almost all t ≥ 0, and for 0 < r < 5/2,
d
kU k2V1 + kΘt k2M0ε + ωkZk2X2 + δkΘt k2M0ε
dt
(3.41)
2(2r−1)
≤ C(kU k2X2 + 1) + Cω kuk2H 1 (Γ) (kuk2H 1 (Γ) kukL2 (Γ) ).
ωαhu, ziL2 (Ω) ≤ ωα2 kuk2L2 (Ω) +
On the left-hand side, we estimate the term ωkZk2X2 using
2
2
kU k2V1 = hU, Aα,β
W U iX2 = hU, ZiX2 ≤ Cω kU kX2 + ωkZkX2 .
(3.42)
Finally, with (3.42) and the uniform bounds (3.16), we now obtain from (3.41),
with m1 := min{1, δ} > 0,
d
kU k2V1 + kΘt k2M0ε + m1 kU k2V1 + kΘt k2M0ε
dt
(3.43)
≤ Cω (1 + kuk2H 1 (Γ) )(kU k2V1 + kΘt k2M0ε ) + C,
where Cω > 0 depends on Pe0 from (3.16). Now from (3.9), we immediately find the
following dissipation integral
Z t+1
ω
kU (τ )k2V1 dτ ≤ C,
(3.44)
t
and we may apply a Grönwall-type inequality (see e.g. Proposition 4.3 below) to
(3.43). We also recall (1.40) yields, for some C∗ > 0,
t 2
t 2
t 2
C∗−1 kΦt k2V2 ≤ kAα,β
W Φ kM0ε = kΘ kM0ε ≤ C∗ kΦ kV2 .
(3.45)
Hence, there are constants M1 ≥ 1 and P1 > 0, both uniform in t, such that for all
t ≥ 0, (3.43) produces, for all t ≥ 0,
kU (t)k2V1 + kΦt k2M2ε ≤ M1 e−m1 t (kU0 k2V1 + kΦ0 k2M2ε ) + P1
≤ M1 Re−m1 t + P1 ,
(3.46)
20
J. L. SHOMBERG
EJDE-2016/47
where the last inequality follows because kΦ0 kM2ε ≤ kΦ0 kKε2 ≤ R.
To show (3.19) holds we need to control the last two terms of the norm (1.35).
First, it is easy to see from (3.46) that for all t ≥ 0
kU (t)k2V1 ≤ kU (t)k2V1 + kΦt k2M2ε ≤ M1 R + P1 .
(3.47)
Then the conclusions of Lemmas 1.7 and 1.9 given above now take the form
εkTε Φt k2M0ε + sup τ Tε (τ ; Φt )
τ ≥0
≤ e−δt εkTε Φ0 k2M0ε + 2 sup τ Tε (τ ; Φ0 )(t + 2) + M1 Re−m1 t + P1
τ ≥1
≤e
−m1 t
(3.48)
(R(M1 + 1) + Q(R)(t + 1)) + P1
≤ Q(R)e−m1 t (t + 1) + P1 .
Together, the estimates (3.46) and (3.48) show that (3.19) holds.
The existence of a bounded set Bε1 in Vε1 that is absorbing and positively invariant
for Sε (t) follows from (3.19). Indeed, define
p
Bε1 := (U, Φ) ∈ Vε1 : k(U, Φ)kVε1 ≤ 2P1 + 1 .
Then, given any nonempty bounded subset B in Hε0 \Bε1 , and after possibly enlarging
the radius of Bε1 in Hε0 due to the embedding Vε1 ,→ Hε0 , we have that Sε (t)B ⊆ Bε1 ,
in Hε0 , for all t ≥ t1 where t1 = t1 (R) ≥ 0 is such that there holds
1
e− min{δ,1}t1 (t1 + 1) ≤
.
(3.49)
Q(R)
This establishes (C1) and completes the proof when ε ∈ (0, 1].
The following result refers to the strong solutions developed in [17, Theorem
3.11] (see Theorem 2.6 above) whose initial data is now taken in Vε1 ⊂ Hε0 .
Corollary 3.10. For all Υ = (U0 , Φ0 ) ∈ Hε1 = V1 × M2ε , it follows that any strong
solution (U (t), Φt ) to Problem (1.1) is bounded, uniformly in t and ε; indeed, thanks
to (3.19) there is a constant Pe1 > 0, depending on the bound P1 and the initial
datum, but independent of t and ε, in which,
lim sup kSε (t)(U0 , Φ0 )kV 1 ≤ Pe1 .
(3.50)
ε
t→+∞
We can now give a decay estimate for Φt in M1ε .
Lemma 3.11. There holds, for all ε ∈ (0, 1], ω ∈ (0, 1), Υ0 = (U0 , Φ0 ) ∈ Vε1 , and
for all t ≥ 0,
kΦt k2M1ε ≤ kΦ0 k2M1ε e−δt/2ε + C(Pe0 )ε.
(3.51)
Proof. Let ε ∈ (0, 1], ω ∈ (0, 1) and Υ0 = (U0 , Φ0 ) ∈ Hε0 . As in the proof of Lemma
3.2, take Π = Φt (s) in equation (2.9) to obtain
Z ∞
t
µε (s)h∂t Φt (s), Aα,β
W Φ (s)iX2 ds
0
Z ∞
Z ∞
t
t
2
=
µε (s)hTε Φt (s), Aα,β
Φ
(s)i
ds
+
µε (s)hU, Aα,β
X
W
W Φ (s)iX2 ds.
0
0
Combining (1.34), (3.5), (3.8), and (3.10), we obtain
1 d
δ
kΦt k2M1ε + kΦt k2M1ε ≤ hU, Φt iM1ε .
2 dt
2ε
(3.52)
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
Estimating the product on the right-hand side with Young’s inequality,
Z ∞
hU, Φt iM1ε =
µε (s)hU, Φt iV1 ds
Z0 ∞
≤
µε (s)kU kV1 kΦt kV1 ds
0
21
(3.53)
≤ kU kV1 kΦt kM1ε
1
δ
kU k2V1 + kΦt k2M1ε ,
δ
4ε
we combine (3.52) and (3.53) to find that, for almost all t ≥ 0,
d
δ
1
(3.54)
kΦt k2M1ε + kΦt k2M1ε ≤ kU k2V1 .
dt
4ε
δ
Thus, applying a Grönwall type inequality whereby integrating (3.54) over the
interval (0, t), recalling the uniform bound (3.50), produces (3.51).
≤
Corollary 3.12. From Lemma 3.11 we obtain the limit, for each t > 0 fixed,
lim kΦt kM1ε = 0.
ε→0
(3.55)
In addition, since e−δt/2ε < e−δt/2 εδt/2 < εδT /2 for all ε ∈ (0, 1] and for all t in the
compact interval [0, T ], for some T > 0, then inequality (3.51) is estimated by,
n
o
kΦt k2M1ε ≤ max kΦ0 k2M1ε , C(Pe0 ) (eδT /2 + ε).
1/2
1
Define the constants Λ0 = max kΦ0 k2M1 e−δT /2 , C(Pe0 )
and p0 = min{ δT
4 , 2 }.
ε
Then, for all ε ∈ (0, 1] and for all t ∈ [0, T ], there holds
kΦt kM1ε ≤ Λ0 εp0 .
We now go on to establish the next condition of Proposition 3.8.
Lemma 3.13. Condition (C2) holds for each ε ∈ (0, 1] and ω ∈ (0, 1). The constants t∗ and `∗ depend on ω, δ and the constant due to the embedding V1 ,→ X2 .
Proof. Let ε ∈ (0, 1] and ω ∈ (0, 1). Let Υ0 = (U0 , Φ0 ), Ξ0 = (V0 , Ψ0 ) ∈ Bε1 .
Define the pair of trajectories, for t ≥ 0, Υ(t) = Sε (t)Υ0 = (U (t), Φt ) and Ξ(t) =
Sε (t)Ξ0 = (V (t), Ψt ). For each t ≥ 0, decompose the difference ∆(t) := Υ(t) − Ξ(t)
with ∆0 := Υ0 − Ξ0 as follows:
b + Ξ(t)
b
∆(t) = Υ(t)
b = (W
c (t), Θ
b t ) are solutions of the problems:
b
b t ) and Ξ(t)
where Υ(t)
= (Vb (t), Ψ
Z ∞
b
bt
∂t Vb (t) + ωA0,β
µε (s)Aα,β
W V (t) +
W Ψ (s) ds = 0,
0
b t (s) = Tε Ψ
b t (s) + Vb (t),
∂t Ψ
(3.56)
b
Υ(0)
= Υ0 − Ξ0 ,
and
c (t) + ωA0,β W
c (t) +
∂t W
W
Z
0
∞
bt
µε (s)Aα,β
W Θ (s) ds + F (U (t)) − F (V (t)) = 0,
b t (s) = Tε Θ
b t (s) + W
c (t),
∂t Θ
b
Ξ(0)
= 0.
(3.57)
22
J. L. SHOMBERG
EJDE-2016/47
Step 1. (Proof of (3.17).) By estimating along the usual lines, after multiply0
bt
ing (3.56)1 by Vb in X2 and multiplying equation (3.56)2 by Aα,β
W Ψ in Mε =
2
2
Lµε (R+ ; X ), we easily obtain the differential inequality,
1 d b 2
b t k2 1 + C −1 ωkVb k2 2 + δ kΨ
b t k2 1 ≤ 0,
kV kX2 + kΨ
Mε
X
Mε
Ω
2 dt
2
(3.58)
where the constant CΩ > 0 is due to the embedding V1 ,→ X2 ; i.e., kVb k2X2 ≤
CΩ kVb k2V1 . Set m2 := min{2CΩ−1 ω, δ} > 0. Thus, (3.58) becomes, for almost all
t ≥ 0,
d b 2
b t k2 1 + m2 (kVb k2 2 + kΨ
b t k2 1 ) ≤ 0.
kV kX2 + kΨ
Mε
X
Mε
dt
After applying a Grönwall inequality, we have that for all t ≥ 0,
b t )kH0 ≤ k∆0 kH0 e−m2 t/2 .
k(Vb (t), Ψ
ε
ε
(3.59)
Set t∗ := max{t1 , m22 ln 4} (recall t1 was defined in (3.49) in the proof of Lemma
b ∗ ) = (Vb (t∗ ), Φ
b t∗ ), and
3.9). Then, for all t ≥ t∗ , (3.17) holds with L = Υ(t
∗
`∗ = e−m2 t
/2
<
1
.
2
Before we show that (3.18) holds, we need to establish a crucial bound.
c and Θ
b t .) We claim, for each 0 < T < ∞,
Step 2. (A preliminary bound for W
there holds
c ∈ L∞ ([0, T ]; X2 ) ∩ L2 ([0, T ]; V1 ),
W
(3.60)
b t ∈ L∞ ([0, T ]; M1ε ).
Θ
(3.61)
c in X2 and multiply equation
To show this, we multiply equation (3.57)1 by W
α,β b t
0
(3.57)2 by AW Θ in Mε . Summing the resulting two identities produces,
1 d c 2
c, W
c iX2 − hTε Θ
b t, Θ
b t iM1
b t k2 1 + ωhA0,β W
kW kX2 + kΘ
Mε
W
ε
2 dt
c
+ hF (U ) − F (V ), W iX2
(3.62)
= 0.
The first of the three products above can be re-written, using the definition of the
V1 norm (see (1.18)), as
α,β c c
c c
ωhA0,β
b wi
b L2 (Ω)
W W , W iX2 = ωhAW W , W iX2 − ωαhw,
c k2 1 − ωαkwk
= ωkW
b 2L2 (Ω)
V
= ω k∇wk
b 2L2 (Ω) + k∇Γ wk
b 2L2 (Γ) + βkwk
b 2L2 (Γ) .
(3.63)
As with the above estimate (3.28), we have
b t, Θ
b t iM1 ≤ − δ kΘ
b t k2 1 .
hTε Θ
Mε
ε
2
(3.64)
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
23
Using assumptions (1.41) and (1.42) with data in the bounded set Bε1 and the
uniform bound (3.16), we now estimate the nonlinear terms as follows
hf (u) − f (v), wi
b L2 (Ω) ≤ k(f (u) − f (v))wk
b L1 (Ω)
≤ kf (u) − f (v)kL6/5 (Ω) kwk
b L6 (Ω)
b L6 (Ω)
≤ `1 k(u − v)(1 + |u − v|r1 )kL6/5 (Ω) kwk
(3.65)
≤ `1 ku − vkL6 (Ω) (1 + ku − vkrL13r1 /2 (Ω) )kwk
b L6 (Ω)
≤ Ckwk
b H 1 (Ω) ,
where C = C(`1 , Ω, Pe0 , r1 ) > 0 and the last inequality follows from the fact that
H 1 (Ω) ,→ L6 (Ω) and H 1 (Ω) ,→ L3r1 /2 (Ω) because 1 ≤ r1 < 52 . Similarly for ge
(here the estimate is easier because H 1 (Γ) ,→ Lp (Γ) for 1 ≤ p < ∞ as Γ is two
dimensional),
he
g (u) − ge(v), wi
b L2 (Γ) ≤ Ckwk
b H 1 (Γ) .
(3.66)
Thus, (3.65) and (3.66) show that
c k2 1 ,
c iX2 ≤ Cω k∆0 k2 0 + ω kW
hF (U ) − F (V ), W
(3.67)
Hε
V
2
where Cω ∼ C
ω . Together (3.62)-(3.67) yields the differential inequality, which holds
for almost all t ≥ 0,
d c 2
b t k2 1
kW kX2 + kΘ
Mε
dt
(3.68)
b t k2 1
+ ω(k∇wk
b 2L2 (Ω) + k∇Γ wk
b 2L2 (Γ) + βkwk
b 2L2 (Γ) ) + δkΘ
Mε
≤ Cω k∆0 k2H0ε .
Now integrating (3.68) with respect to t in [0, T ], for some fixed 0 < T < ∞, we
obtain
Z t 2
t 2
c
b
kW (t)k 2 + kΘ k 1 +
ω k∇w(τ
b )k2 2
X
Mε
L (Ω)
0
b τ k2 1 dτ
+ k∇Γ w(τ
b )k2L2 (Γ) + βkw(τ
b )k2L2 (Γ) + δkΘ
Mε
(3.69)
≤ Cω k∆0 k2H0ε T.
Using (3.69), we easily deduce the claim (3.60)-(3.61).
c
Step 3. (Proof of (3.18)) We begin by multiplying equation (3.57)1 by K = Aα,β
W W
α,β
2
in X , then, after applying AW to equation (3.57)2 , we multiply the result by
0
2
2
bt
Λt = Aα,β
W Θ in Mε = Lµε (R+ ; X ). This leaves us with the two identities,
c , KiX2 + ωhA0,β W
c , KiX2 + hAα,β Θ
b t , KiM0
h∂t W
W
W
ε
+ hF (U ) − F (V ), KiX2 = 0.
(3.70)
and
α,β
α,β c
t
bt t
bt t
h∂t Aα,β
W Θ , Λ iM0ε = hAW Tε Θ , Λ iM0ε + hAW W , Λ iM0ε .
Observe that
α,β c
t
bt
hAα,β
W Θ , KiM0ε = hΛ , AW W iM0ε .
Hence, combining (3.70) and (3.71) through (3.72),
c , KiX2 + ωhA0,β W
c , KiX2 + h∂t Aα,β Θ
b t , Λt iM0
h∂t W
W
W
ε
bt t
− hAα,β
W Tε Θ , Λ iM0ε + hF (U ) − F (V ), KiX2 = 0.
(3.71)
(3.72)
(3.73)
24
J. L. SHOMBERG
EJDE-2016/47
The first three products can be re-written as follows,
1 d c 2
kW kV1 ,
2 dt
− ωαhw,
b kiL2 (Ω)
c , KiX2 = h∂t W
c , Aα,β W
c iX2 = h∂t W
c, W
c iV1 =
h∂t W
W
α,β c
c
ωhA0,β
W W , KiX2 = ωhAW W , KiX2
= ωkKk2X2 − ωαhw,
b kiL2 (Ω) ,
(3.74)
(3.75)
1 d t 2
kΛ kM0ε .
(3.76)
2 dt
Inserting (3.74)-(3.76) into (3.73) gives us the differential identity
1 d c 2
b t , Λt iM0 + hF (U ) − F (V ), KiX2
kW kV1 + kΛt k2M0ε + ωkKk2X2 − hTε Θ
ε
2 dt
= ωαhw,
b kiL2 (Ω) .
(3.77)
Similar to (3.28), we have
t
t
bt t
h∂t Aα,β
W Θ , Λ iM0ε = h∂t Λ , Λ iM0ε =
δ t 2
bt t
hAα,β
(3.78)
W Tε Θ , Λ iM0ε ≤ − kΛ kM0ε ,
2
and in a similar fashion to (3.67), we find
ω
|hF (U ) − F (V ), KiX2 | ≤ Cω k∆0 k2H0ε + kKk2V1 ,
(3.79)
2
where Cω ∼ C
ω . We also estimate
c k2 2 + ω kKk2 2
ωαhw,
b kiL2 (Ω) ≤ ωα2 kW
(3.80)
X
X
4
Together (3.77)-(3.80) yields the differential inequality, which holds for almost all
t ≥ 0,
d c 2
t 2
c 2
kW kV1 + kΛt k2M0ε + ωkAα,β
W W kX2 + δkΛ kM0ε
dt
(3.81)
c k2 2 + Cω k∆0 k2 0 .
≤ αkW
X
Hε
Now integrating (3.81) with respect to t in [0, T ], for some fixed 0 < T < ∞, we
obtain
Z t
2
t 2
2
τ 2
c
c
kW (t)kV1 + kΛ kM0ε +
(ωkA0,β
(3.82)
W W (τ )kX2 + δkΛ kM0ε )dτ
0
Z t
c (τ )k2 2 dτ + Cω k∆0 k2 0 T,
(3.83)
≤
αkW
Hε
X
0
where the right-hand side of the inequality makes sense thanks to (3.60). Now
omitting the second and third terms from the left-hand side of (3.83), the following
bound follows easily with Grönwall’s inequality
c (t)k2 1 ≤ Cω k∆0 k2 0 T eαT ,
kW
V
Hε
(3.84)
kΛt k2M0ε ≤ Cω k∆0 k2H0ε T eαT ,
(3.85)
and with this
also follows.
To obtain the desired bound from (3.84) and (3.85), first recall that there is
C∗ > 0 (cf. (1.40)) such that
−1 b t 2
bt 2
kΛt k2M0ε = kAα,β
W Θ kM0ε ≥ C∗ kΘ kM2ε .
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
25
Thus, letting T = t∗ (from Step 1), we obtain, for some positive monotonically
increasing function M2 (·),
c (t∗ ), Θ
b t∗ )kH1 ≤ M2 (t∗ )k∆0 kH0 .
k(W
ε
ε
(3.86)
Now it suffices to show that for some positive constant C(T ), there holds for all
t ∈ [0, T ],
b t k2 2 ≤ C(T )k∆0 k2 0 .
kΘ
Hε
Kε
(3.87)
First, we see that with an application of Lemma 1.10 with (3.84) and 3.85 there
holds, for all t ∈ [0, T ],
b t ) ≤ C(T )k∆0 k2 0 ,
sup τ Tε (τ ; Θ
Hε
(3.88)
τ ≥1
and secondly, by applying the weak form of Lemma 1.7 (see Remark 1.8), we find
that for all t ∈ [0, T ],
εkTε Φt k2M0ε ≤ C(T )k∆0 k2H0ε .
(3.89)
Together (3.88)-(3.89) establish (3.87). Therefore, inequality (3.18) now follows
c (t∗ ), Θ
b t∗ ) and ℘∗ = M2 (t∗ ) ≥ 0 (for a suitably updated
with R = Ξ(t∗ ) = (W
function M2 ). This completes the proof of (C2).
Lemma 3.14. Condition (C3) holds for each ε ∈ (0, 1] and ω ∈ (0, 1).
Proof. Let ε ∈ (0, 1] and ω ∈ (0, 1). Let R > 0 and Υ0 = (U0 , Φ0 ) ∈ Vε1 where
kΥ0 kVε1 ≤ R. Directly from (3.50), there holds,
kSε (t)Υ0 kVε1 ≤ Pe1 ,
but where now the size of the initial data, R, depends on the size of Bε1 . Hence, on
the compact interval [t∗ , 2t∗ ], the map t 7→ S(t)Υ0 is Lipschitz continuous for each
fixed Υ0 ∈ Bε1 . This means there is a constant L = L(t∗ ) > 0 such that
kSε (t1 )Υ0 − Sε (t2 )Υ0 kH0ε ≤ L|t1 − t2 |.
Together with the continuous dependence estimate (2.15), (C3) follows.
Remark 3.15. According to Proposition 3.8, for each ε ∈ (0, 1], the semigroup
Sε (t) : Hε0 → Hε0 possesses an exponential attractor, Mε ⊂ Bε1 , which attracts
bounded subsets of Bε1 exponentially fast (in the topology of Hε0 ). Moreover, in light
of the results in this section – which are uniform in the perturbation parameter ε
– we now simply accept the corresponding results for the simpler limit Problem
(1.2). In this setting we use the notation for the compact absorbing set B01 and the
exponential attractor M0 admitted by the semigroup S0 (t) : H00 = X2 → X2 .
Remark 3.16. To show that the attraction property (iii) in Theorem 3.6 also holds
– that is, to show that the basin of attraction of Mε is all of Hε0 – we appeal to
the transitivity of the exponential attraction in Proposition 4.2 and Theorem 3.17
below.
26
J. L. SHOMBERG
EJDE-2016/47
3.3. Basin of attraction (and global attractors). The main result in this section has two purposes: primary, per the above remark, it will help us show that
the exponential attractors we seek attract every bounded subset in Hε0 (not just
Bε1 ). This property is sometimes not obvious because of the difficulties using spaces
involving memory (we refer the reader to Section 1 of this article and to the rate
of attraction of Bε1 as found in Lemma 3.9). However, we overcome this problem,
partly, by proving a condition on the solution semigroup Sε that is also essential for
the existence of global attractors (also called a universal attractors); we refer to the
asymptotic compactness/smoothing of Sε , which happens to occur in our case with
an exponential rate. Together, the asymptotic compactness os Sε (Theorem 3.17
below) and the existence of an absorbing sets in Hε0 (Lemma 3.2) will guarantee
the existence of a global attractor that is compact in Hε0 and bounded in Vε1 .
Theorem 3.17. For each ε ∈ [0, 1], there is a positive constant %1 and a monotonically increasing function Q(·) in which for every nonempty bounded subset B
of Hε0 there holds, for all t ≥ 0,
distH0ε (Sε (t)B, Bε1 ) ≤ Q(kBkH0ε )e−%1 t .
Proof. Because of the smoothing properties of the associated with the Wentzell
parabolic Problem (1.2) (cf. [13]), we limit ourselves to the case when ε ∈ (0, 1].
Let ε ∈ (0, 1] and B be a nonempty bounded subset of Hε0 . By recalling Lemma
3.2, we already know that there is a bounded absorbing set that is exponentially
attracting in Hε0 , i.e., for all t ≥ 0 there holds
distH0ε (Sε (t)B, Bε0 ) ≤ Q(kBkH0ε )e−ν0 t ,
so owing once again to the transitivity of exponential attraction (cf. Proposition
4.2 below) it suffices to show that, for all t ≥ 0,
distH0ε (Sε (t)Bε0 , Bε1 ) ≤ Q(P0 )e−%0 t ,
(3.90)
for some positive constant %0 and for some positive monotonically
increasing func√
tion Q(·), each independent of ε. (Recall from (3.3) that P0 + 1 is the radius of
Bε0 .)
To prove (3.90), the idea is to show that for each ε ∈ (0, 1] and for each Υ0 ∈ Hε0
we can decompose the semigroup
Sε (t)Υ0 = Zε (t)Υ0 + Kε (t)Υ0
where the operators Zε are uniformly (exponentially) decaying to zero and Kε
are uniformly compact (bounded in Vε1 ) for large t. This is done in the following
lemmas.
The following decomposition and subsequently more general lemmas, as we will
allow the datum to belong to any bounded subset of the phase space Hε0 , can be
seen to follow [7, Theorem 6.10–Lemma 6.12] with obvious changes to account for
the dynamic boundary conditions with memory. Hence, we will limit the proofs to
sketches of the most important details.
First, choose a constant MF > 0, based on (1.43), (1.44), and (2.1), so that the
map defined by, for all s ∈ R,
F0 (s) := F (s) + MF s,
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BOUNDARY CONDITIONS WITH MEMORY
27
satisfies, for every s ∈ R,
F00 (s)
0
≥
.
0
(3.91)
Next, let Υ0 = (U0 , Φ0 ) ∈ Hε0 . Then rewrite Problem (1.1) into the system of
equations in (V, Ψ) and (W, Θ), where (V, Ψ) + (W, Θ) = (U, Φ),
Z ∞
t
∂t V (t) + ωA0,β
V
(t)
+
µε (s)Aα,β
W
W Ψ (s) ds + F0 (U (t)) − F0 (W (t)) = 0,
0
(3.92)
∂t Ψt (s) = Tε Ψt (s) + V (t),
(V (0), Ψ0 ) = Υ0 ,
and
∂t W (t) + ωA0,β
W W (t) +
∞
Z
0
t
µε (s)Aα,β
W Θ (s)ds + F0 (W (t)) − MF U (t) = 0,
∂t Θt (s) = Tε Θt (s) + W (t),
(3.93)
0
(W (0), Θ ) = 0.
In view of Lemmas 3.18 and 3.19 below, we define the one-parameter family of
maps, Kε (t) : Hε0 → Hε0 , by
Kε (t)Υ0 := (W (t), Θt ),
where (W, Θ) is a solution of (3.93). With such (W, Θ), we may define a second
function (V, Ψ) as the solution of (3.92). Through the dependence of (V, Ψ) on
(W, Θ) and (U (0), Φ0 ) = Υ0 , the solution of (3.92) defines a one-parameter family
of maps, Zε (t) : Hε0 → Hε0 , defined by
Zε (t)Υ0 := (V (t), Ψt ).
Notice that if (V, Ψ) and (W, Θ) are solutions to (3.92) and (3.93), respectively,
then the function (U (t), Φt ) := (V (t), Ψt ) + (W (t), Θt ) is a solution to Problem
(1.1).
The next result shows that the operators Zε are uniformly decaying to zero in
Hε .
Lemma 3.18. For each ε ∈ (0, 1] and Υ0 = (U0 , Φ0 ) ∈ Hε0 , there exists a unique
global weak solution (V, Ψ) ∈ C([0, ∞); Hε0 ) to problem (3.92). Moreover, given
R > 0, then for all Υ0 ∈ Hε0 with kΥ0 kH0ε ≤ R for all ε ∈ (0, 1], there exists ν00 > 0,
independent of ε, such that, for all t ≥ 0,
0
kZε (t)Υ0 kH0ε ≤ Q(R)e−ν0 t .
(3.94)
Proof. The existence of a global weak solution to (3.92) follows as the proof of [18,
Theorem 2.5]. It remains to show that (3.94) holds.
The proof is very similar to the proof of Lemma 3.2 save that the assumptions
(1.43)-(1.44) become crucial. Indeed, the constant C on the right-hand side of
(3.13) vanishes because nonlinear terms now satisfy the bound
hF0 (U ) − F0 (W ), V iX2 ≥ 0
as here V = U − W and (3.91) holds.
The following lemma establishes the uniform compactness of the operators Kε .
28
J. L. SHOMBERG
EJDE-2016/47
Lemma 3.19. For each ε ∈ (0, 1] and Υ0 = (U0 , Φ0 ) ∈ Hε0 , there exists a unique
global weak solution (W, Θ) ∈ C([0, ∞); Hε0 ) to problem (3.93). Moreover, given
R > 0, then for all Υ0 ∈ Hε0 with kΥ0 kH0ε ≤ R for all ε ∈ (0, 1], there holds for all
t ≥ 0,
kKε (t)Υ0 kVε1 ≤ Q(R),
Furthermore, the operators Kε are uniformly compact in Hε0 .
Proof. Again, in light of [18, Theorem 2.5], it remains to show that the operators
Kε are uniformly compact in Hε0 .
This time we appeal to Lemma 3.9 whereby only trivial changes are required in
the proof in order to show Lemma 3.19 holds.
Remark 3.20. These results – with datum contained to the absorbing set Bε0
– complete the proof of Theorem 3.17. Consequently, the existence of a (finite
dimensional) global attractor Aε , ε ∈ (0, 1], for Sε follows.
Theorem 3.21. For each ε ∈ (0, 1], the semigroup Sε admits a unique global
attractor
Aε = ω(Bε0 ) := ∩s≥0 ∪t≥s Sε (t)Bε0
H0ε
in Hε0 . Moreover, the following statements hold:
(1) For each t ≥ 0, Sε (t)Aε = Aε , and
(2) For every nonempty bounded subset B of Hε0 ,
lim distH0ε (Sε (t)B, Aε ) = 0.
t→∞
(3.95)
(3) The global attractor Aε is bounded in Vε1 (hence, compact in Hε0 ) and trajectories on Aε are strong solutions (in the sense of Definitions 2.4).
(4) The fractal dimension is bounded, uniformly in ε, i.e.,
dimF (Aε , Hε0 ) ≤ dimF (Mε , Hε0 ) ≤ C < ∞,
for some constant C > 0 independent of ε.
Proof. The existence and boundedness of the global attractor for Problem (1.2)
can be found in [12, Theorem 2.3] and the references therein. Thus, it suffices to
show the result for the perturbation Problem (1.1), with ε ∈ (0, 1]. By referring to
the standard literature (cf. e.g. [1, 39]) and Lemma 3.2, Lemma 3.9, and Theorem
3.17, the proof is complete.
3.4. Robustness and Hölder continuity of the exponential attractors.
What remains in this section is to show that the family of exponential attractors is robust, or Hölder continuous with respect to the perturbation parameter ε.
As a preliminary step, we follow, for example [6, see p. 177] among others, and
define the so-called canonical extension map, E : X2 → Mε , by
E(U ) = 0.
(3.96)
With this, define the lift mapping, LX2 → Hε0 , by
L(U ) = (U, E(U )) = (U, 0).
(3.97)
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BOUNDARY CONDITIONS WITH MEMORY
29
Theorem 3.22. Let the assumptions of Theorem 3.6 be satisfied. For each ε, the
semigroup of solution operators, Sε (t) admits an exponential attractor Mε in which
the family of compact sets (Mε )ε∈[0,1] defined by
(
LM0 for ε = 0
Mε :=
(3.98)
Mε
for ε ∈ (0, 1]
is Hölder continuous for every ε ∈ [0, 1], i.e., there exist constants Λ > 0, τ ∈
(0, 1/2] independent of ε, such that, for every 0 ≤ ε2 < ε1 ≤ 1, the symmetric
Hausdorff distance satisfies
τ
distsym
H0 (Mε1 , Mε2 ) ≤ Λ(ε1 − ε2 ) .
ε1
(3.99)
Remark 3.23. The symmetric Hausdorff distance between two subsets A, B of a
Banach space X is defined as
distsym
X (A, B) := max{distX (A, B), distX (B, A)}.
(3.100)
More precisely, the condition given in (3.99) implies the family of attractors is both
upper- and lower-semicontinuous (thus, continuous) at each value of the perturbation parameter ε ∈ [0, 1).
To prove Theorem 3.22 we develop the main assumptions of the abstract results
found in the seminal works [20, 21]. As in Proposition 3.8 above, the assumptions
suited specifically for our needs appear in [7, (H2) and (H3) of Theorem A.2].
As above, the number L > 0 shown below is used to denote the (local) Lipschitz
constant of the mapping F : V1 → X2 .
Proposition 3.24. Let the assumptions of Proposition 3.8 be satisfied. In addition,
assume the following:
(C4) The canonical extension map E|B01 : X2 → Hε0 given by (3.96) is Lipschitz
continuous.
(C5) There is a constant Λ1 = Λ1 (L, Ω, t∗ ) > 0 such that, for all t ∈ [t∗ , 2t∗ ] and
for all Υ0 = (U0 , Φ0 ) ∈ Bε1 ,
√
(3.101)
kSε (t)Υ0 − LS 0 (t)PΥ0 kH0ε ≤ Λ1 ε.
Here, P : Hε0 → H00 denotes the projection defined by, for all Υ = (U, Φ) ∈
Hε0 ,
PΥ = U.
(C6) There is a constant Λ2 = Λ2 (L, Ω, t∗ ) > 0 such that, for all t ∈ [t∗ , 2t∗ ],
Υ0 = (U0 , Φ0 ) ∈ Bε11 ⊂ Hε11 and, for all 0 < ε2 < ε1 ≤ 1,
kSε1 (t)Υ0 − Sε2 (t)Υ0 kH0ε ≤ Λ2 (ε1 − ε2 )1/2 .
1
(3.102)
Then, the family of exponential attractors (Mε )ε∈[0,1] is Hölder continuous for
every ε ∈ [0, 1] in the sense of Theorem 3.22.
Remark 3.25. Condition (C6) below does not appear in [7], but rather we now
borrow [21, (H7) of Theorem 4.4], cf. also [32, (P4) of Theorem 2.1].
Lemma 3.26. Condition (C4) holds.
Proof. Based on the definition of E given in (3.96), the result is vacuously true. 30
J. L. SHOMBERG
EJDE-2016/47
The following lemma proves condition (C5) of Proposition 3.24. It shows that
the difference between the semigroups Sε (t) and the lifted limit semigroup LS 0 (t)
in Hε0 , on finite time intervals, is of order ε1/2 .
Lemma 3.27. Let T > 0. For all ε ∈ (0, 1], ω ∈ (0, 1) and Υ0 = (U0 , Φ0 ) ∈ Hε0
such that kΥ0 kH0ε ≤ R for all ε ∈ (0, 1], there exists a positive constant C(T ),
independent of ε, but depending on ω and T , in which, for all t ≥ [0, T ],
kSε (t)Υ0 − LS 0 (t)PΥ0 kH0ε ≤ C(T )ε1/2 .
(3.103)
b
b (t), Φ
b t ) denote the solution of Problem Pε corresponding to
Proof. Let Υ(t)
= (U
the initial data Υ0 = (U0 , Φ0 ) ∈ Bε1 and let U (t) denote the solution of Problem P0
corresponding to the initial data PΥ0 = U0 ∈ B01 . With the solution U (t), define
the function Φt by the solution to the Cauchy problem
∂t Φt = Tε Φt + U (t)
Φ0 = QΥ0 = Φ0 ∈ M0ε .
(3.104)
With the (unique) solution to (3.104) (cf. Corollary 1.5), define Υ(t) := (U (t), Φt )
for all t ≥ 0. Let
b
b − Υ(t)
∆(t)
= (Z(t), Θt ) : = Υ(t)
b (t), Φ
b t ) − (U (t), Φt )
= (U
b (t) − U (t), Φ
b t − Φt );
= (U
b
hence, ∆(t)
= (Z(t), Θt ) satisfies the system
Z ∞
t
b
∂t Z(t) + ωA0,β
Z(t)
+
µε (s)Aα,β
W
W Θ (s) ds + F (U (t)) − F (U (t))
0
Z ∞
t
=−
µε (s)Aα,β
W Φ (s) ds,
0
t
(3.105)
t
∂t Θ (s) = Tε Θ (s) + Z(t),
(Z(0), Θ0 ) = 0.
t
2
2
Multiply (3.105)1 by Z in X2 and (3.105)2 by Aα,β
W Θ in Lµε (R+ ; X ), summing the
resulting identities and estimating as in the above arguments, it is not hard to see
that there holds, for almost all t ≥ 0,
1 d
δ
kZk2X2 + kΘt k2M1ε + ωkZk2V1 + kΘt k2M1ε
2 dt
2ε Z
(3.106)
∞
α,β t
2
b ) − F (U ), ZiX2 + ωαkzk 2
2
≤ −hF (U
−
µ
(s)hA
Φ
(s),
Zi
ds.
ε
X
L (Ω)
W
0
Recall that with (1.43) we obtain
b ) − F (U ), ZiX2 ≤ MF kZk2 2 .
− hF (U
X
(3.107)
For the remaining term on the right-hand side, we apply the definition of the norm
and Young’s inequality to find
Z ∞
Z ∞
α,β t
−
µε (s)hAW Φ (s), ZiX2 ds = −
µε (s)hΦt (s), ZiV1 ds
0
0
≤ kZkV1 kΦt kM1ε
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BOUNDARY CONDITIONS WITH MEMORY
≤ ωkZk2V1 +
31
1
kΦt k2M1ε .
4ω
Recall that, by (3.51), for all t ≥ 0,
kΦt k2M1ε ≤ kΦ0 k2M1ε e−δt/2ε + Cε,
(3.108)
where C > 0 depends on the bound Pe0 , but is uniform in ε and t. Collecting
(3.106)-(3.108) yields,
d
kZk2X2 + kΘt k2M1ε + δkΘt k2M1ε
dt
(3.109)
≤ 2(ωα + MF )kZk2X2 + CεkZk2X2 + Cω kΦ0 k2M1ε e−δt/2ε + ε .
Integrating (3.109) with respect to t on the interval [0, T ], for T > 0, and then
applying the initial conditions (3.106)3 , as well as the uniform bound (3.16), we
have
Z
t
kZ(t)k2X2 + kΘt k2M1ε ≤
0
CkZ(τ )k2X2 dτ + C(T )ε.
(3.110)
Next we seek an appropriate bound on the term with Z. It follows from (3.110)
and Gronwall’s inequality that there holds, for all t ≥ 0 and for all ε ∈ (0, 1],
kZ(t)k2X2 ≤ C(T )ε,
(3.111)
where C > 0 depends on ω, δ, and of course T , but not ε.
√
Returning to (3.110), we now see that there holds, for all t ∈ [ ε, T ] and for all
ε ∈ (0, 1],
k(Z(t), Θt )k2H0ε ≤ C(T )ε.
(3.112)
Therefore (3.103) follows. This finishes the proof.
We will establish the Hölder continuity with the following lemma. With regard
to [21], in particular, hypothesis (H7) of Theorem 4.4 there, we do not perform an
ε-scaling of the memory variable.
Lemma 3.28. Condition (C6) holds.
e
e (t), Φ
e t)
Proof. Assume 0 < ε2 < ε1 ≤ 1. Let Υ0 = (U0 , Φ0 ) ∈ B11 . Let Υ(t)
= (U
denote the solution of Problem Pε1 corresponding to the initial datum Υ0 and let
e = (Ve (t), Ψ
e t ) denote the solution Problem Pε corresponding to the same initial
Ξ(t)
2
datum Υ0 . Let
e
e
e t ) : = Υ(t)
e − Ξ(t)
e
∆(t)
= (Z(t),
Θ
e (t), Φ
e t ) − (Ve (t), Ψ
e t)
= (U
e (t) − Ve (t), Φ
et − Ψ
e t ).
= (U
Z ∞
e+
et
e
e
∂t Ze + ωA0,β
Z
µε1 (s)Aα,β
W
W Θ (s) ds + F (U ) − F (V )
0
Z ∞
et
=
(µε2 (s) − µε1 (s))Aα,β
W Ψ (s) ds
0
et
et
e
∂t Θ (s) = Tε1 Θ (s) + Z(t)
e
Z(0)
= 0,
e 0 = 0.
Θ
(3.113)
32
J. L. SHOMBERG
EJDE-2016/47
e t (s) = 0. We proceed in the usual
Observe, by the definition of Tε , (Tε1 − Tε2 )Ψ
2
e in X , and multiplying equation (3.113)2 by
fashion by multiplying (3.113)1 by Z
α,β e t
2
2
AW Θ in Lµε1 (R+ ; X ), summing the results, we arrive at the identity
Z ∞
1 d e 2
e t (s), Aα,β Θ
e t (s)iX2 ds
e t k2 0 + ωkZk
e 21 −
µε1 (s)hTε1 Θ
kZkX2 + kΘ
Mε
V
W
1
2 dt
0
Z ∞
e t (s), Z(t)i
e
=
(µε2 (s) − µε1 (s))hΨ
X2 ds
0
e ) − F (Ve ), Zi
e X2 + ωαkz̃k2 2 .
− hF (U
L (Ω)
(3.114)
We estimate from here along the usual lines to obtain, for almost all t ≥ 0,
Z ∞
e t (s)iX2 ds ≤ δ kΘ
e t (s), Aα,β Θ
e t k2 1 .
µε1 (s)hTε1 Θ
−
(3.115)
Mε
W
1
2ε
1
0
We know there is a constant MF > 0 in which
e ) − F (Ve ), Zi
e X2 ≤ M2 kZk
e 22,
− hF (U
X
e t is uniformly bounded in Bε1 ,
and finally, with the fact that Ψ
1
Z ∞
e t (s), Z(t)i
e
(µε2 (s) − µε1 (s))hΨ
X2 ds
0
Z
ε1 − ε2 ∞
e t (s), Z(t)i
e
µε (s)hΨ
=
X2 ds
ε1 ε2
0
ε1 − ε2 e
e t kM1
kZkX2 kΨ
≤C
ε1
ε2
1 e 2
ε1 − ε2
Q(R1 ) + kZkX2 ,
≤
ε2
2
(3.116)
(3.117)
where R1 > 0 is the radius of the absorbing set Bε11 . After applying (3.115)-(3.117),
we obtain the differential inequality,
d e 2
e t k2 1
kZkX2 + kΘ
Mε
1
dt
e 2 2 + CkΘ
e t k2 1 + ε1 − ε2 Q(R1 ),
≤ 2(M2 + ω + 1)kZk
X
Mε
1
ε2
(3.118)
where M3 := max{2(M2 + ω + 1), C} > 0. We now integrate (3.118) with respect
to t over [0, T ] which in turn yields the Gronwall-type estimate, for all t ∈ [0, T ]
r
ε1 − ε2 Q(R1 )(eM3 T − 1)
t
e
e
k(Z(t), Θ )kH0ε ≤
.
1
ε2
M3
Therefore, (3.102) follows.
Remark 3.29. In conclusion, by Theorem 3.22 the semigroup Sε generated by
the solutions of Problem (1.1) admits a robust family of exponential attractors
(Mε )ε∈[0,1] in Hε0 , Hölder continuous at each ε ∈ [0, 1].
EJDE-2016/47
BOUNDARY CONDITIONS WITH MEMORY
33
4. Appendix
For the reader’s convenience we report some important results that are needed
in this article. The following lemma is from [15, Lemma 2.2]. It is in the spirit of
the H s -elliptic regularity estimate that can be found in [30, Theorem II.5.1].
Lemma 4.1. Consider the linear boundary value problem
−∆u + αu = ψ1 in Ω,
−∆Γ u + ∂n u + βu = ψ2
on Γ.
(4.1)
If (ψ1 , ψ2 )tr ∈ H s (Ω) × H s (Γ), for s ≥ 0 and s + 21 6∈ N, then the following estimate
holds for some constant C > 0,
kukH s+2 (Ω) + kukH s+2 (Γ) ≤ C(kψ1 kH s (Ω) + kψ2 kH s (Γ) ).
(4.2)
The following result is the so-called transitivity property of exponential attraction from [10, Theorem 5.1]).
Proposition 4.2. Let (X , d) be a metric space and let St be a semigroup acting
on this space such that
d(St x1 , St x2 ) ≤ CeKt d(x1 , x2 ),
for appropriate constants C and K. Assume there exists three subsets U1 ,U2 ,U3 ⊂
X such that
distX (St U1 , U2 ) ≤ C1 e−α1 t ,
distX (St U2 , U3 ) ≤ C2 e−α2 t .
Then
0
distX (St U1 , U3 ) ≤ C 0 e−α t ,
α1 α1
.
where C 0 = CC1 + C2 and α0 = K+α
1 +α2
The following statement refers to a frequently used Grönwall-type inequality that
is useful when working with dissipation arguments. We also refer the reader to [5,
Lemma 2.1], [26, Lemma 2.2] and [33, Lemma 5].
Proposition 4.3. Let Λ : R+ → R+ be an absolutely continuous function satisfying
d
Λ(t) + 2ηΛ(t) ≤ h(t)Λ(t) + k,
dt
Rt
where η > 0, k ≥ 0 and s h(τ )dτ ≤ η(t − s) + m, for all t ≥ s ≥ 0 and some
m ≥ 0. Then, for all t ≥ 0,
kem
Λ(t) ≤ Λ(0)em e−ηt +
.
η
Acknowledgments. The author is indebted to the anonymous referees for their
careful reading of the manuscript and for their helpful comments and suggestions.
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Joseph L. Shomberg
Department of Mathematics and Computer Science, Providence College, Providence,
RI 02918, USA
E-mail address: jshomber@providence.edu