TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 361, Number 5, May 2009, Pages 2567–2586
S 0002-9947(08)04789-2
Article electronically published on November 4, 2008
LOCAL WELL POSEDNESS, ASYMPTOTIC BEHAVIOR
AND ASYMPTOTIC BOOTSTRAPPING
FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS
OF THE SECOND ORDER IN TIME
A. N. CARVALHO AND J. W. CHOLEWA
Abstract. A class of semilinear evolution equations of the second order in
time of the form utt + Au + µAut + Autt = f (u) is considered, where −A is the
Dirichlet Laplacian, Ω is a smooth bounded domain in RN and f ∈ C 1 (R, R).
A local well posedness result is proved in the Banach spaces W01,p (Ω)×W01,p (Ω)
when f satisfies appropriate critical growth conditions. In the Hilbert setting,
if f satisfies an additional dissipativeness condition, the nonlinear semigroup
of global solutions is shown to possess a gradient-like attractor. Existence
and regularity of the global attractor are also investigated following the unified semigroup approach, bootstrapping and the interpolation-extrapolation
techniques.
1. Introductory notes
In this article we consider a class of semilinear evolution equations of the second
order in time of the form
(1.1)
utt + Au + µAut + Autt = f (u), t > 0,
with the initial conditions
(1.2)
u(0) = u0 , ut (0) = v0
from a suitably chosen Banach space X 1 . Here −A : D(A) ⊂ X → X is the generator of an exponentially decaying analytic semigroup of bounded linear operators
in X and X 1 is the domain of A with the graph norm.
The equations that fall into this class are known to represent some sort of ‘propagation problems’ (see [4, 6, 22]; also [19] and the references therein), among which
a specific problem is
⎧
⎪
⎨utt − ∆u − µ∆ut − ∆utt = f (u), t > 0, x ∈ Ω,
(1.3)
u = 0, t ≥ 0, x ∈ ∂Ω,
⎪
⎩
u(0, x) = u0 (x), ut (0, x) = v0 (x), x ∈ Ω,
Received by the editors May 21, 2007.
2000 Mathematics Subject Classification. Primary 35G25, 35B33, 35B40, 35B41, 35B65.
Key words and phrases. Evolution equations of the second order in time, existence, uniqueness
and continuous dependence of solutions on initial conditions, asymptotic behavior of solutions,
attractors, regularity, critical exponents.
This research was partially supported by grant # 300.889/92-5 CNPq and grant # 03/10042-0
FAPESP, Brazil.
c
2008
American Mathematical Society
2567
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2568
A. N. CARVALHO AND J. W. CHOLEWA
where µ > 0, Ω is a bounded smooth domain in RN and f : R → R is a locally
Lipschitz function satisfying suitable growth and dissipativeness conditions.
Denote by λ1 the first eigenvalue of −∆ with Dirichlet boundary conditions in
Ω.
In [19] the problem (1.3) was considered in the Hilbert space H01 (Ω) × H01 (Ω).
There the authors prove the existence of a global attractor under the assumptions
lim sup|s|→∞ f (s)
s < λ1 ,
(1.4)
and
|f (s)| ≤ c(1 + |s|ρ ), s ∈ R, where ρ ∈ (1, ∞) is
N +2
if N ≥ 3.
arbitrary if N = 1, 2 and ρ ≤
N −2
(1.5)
N +2
The difficulties connected with the consideration of the critical exponent ρ∗ = N
−2
(N ≥ 3) have been successfully overcome following [14] and [21]. A key point
in this consideration is to obtain the existence of an absorbing set accompanied
by the suitable estimates for the solutions, from which the following asymptotic
compactness property could be inferred:
(1.6)
(ACP) if r > 0, u0n H01 (Ω) +v0n H01 (Ω) ≤ r and tn ∞, there is a subsequence
{nk } so that {u(tnk , u0nk , v0nk )} and {ut (tnk , u0nk , v0nk )} are convergent
in H01 (Ω).
As for the existence-uniqueness results in [19], they have been quoted from previous
references [17, 20], where the proof was based on the Faedo-Galerkin method.
In this paper we show that the semigroup approach, known from the geometric
theory of semilinear problems (see [12, 15]) and dissipative systems theory (see
[10, 13]), is applicable to the problems that fall into the class (1.1)-(1.2) and provides
the relevant information about their solutions.
Within this approach we obtain a local well posedness result for (1.3) in the
Banach space W01,p (Ω) × W01,p (Ω), p ∈ (1, ∞), provided that one of the following
conditions holds:
(H1) p > N and f : R → R is a locally Lipschitz continuous map.
(H2) 1 < p < N and f satisfies
(1.7)
|f (s1 ) − f (s2 )| ≤ c|s1 − s2 |(1 + |s1 |ρ−1 + |s2 |ρ−1 ), s1 , s2 ∈ R,
N +p
.
for some c > 0 and 1 < ρ ≤
N −p
(H3) 1 < p = N and f satisfies (∗) for some c > 0 and ρ ∈ (1, ∞).
(∗)
Within the same approach we then obtain, for p = 2 and assuming (1.4), a global
well posedness result and show that the associated semigroup of global solutions
has a global attractor which has the structure of the attractor of a gradient system
(see [10]).
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SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2569
For p = 2 and in the subcritical case, that is, when
(SC) N = 1 and f : R → R is a locally Lipschitz continuous function,
or N = 2 and (∗) in (1.7) is satisfied with some ρ ∈ (1, ∞), c > 0,
N +2
or N ≥ 3 and (∗) in (1.7) holds with some 1 < ρ <
, c > 0,
N −2
the mentioned approach provides, besides the existence of a global attractor A,
some additional regularity for the solutions in A through an asymptotic bootstrapping procedure.
N +2
In the case involving the critical exponent ρ∗ = N
−2 , and specifically when
(1.8)
(1.9)
(CC)
N ≥ 3, f ∈ C 1 (R, R) and |f (s)| ≤ c(1 + |s| N −2 −1 ), s ∈ R,
N +2
the existence of the global attractor is obtained with the aid of Alekseev’s nonlinear
variation of constants formula (see [2]). In either case the attractor is gradient-like
(see [11]); that is,
A = W u (E),
(1.10)
where E denotes the set of equilibria of (1.3) and W u (E) is the unstable set of E.
To be more specific about our technique and approach note first that in the
variables (t, v), with v = (I + A)u, the problem (1.1)-(1.2) becomes
(1.11)
vtt + µΛvt + Λv = f e (v), t > 0,
with the initial conditions
(1.12)
v(0) = v0 , vt (0) = w0
in a Banach space X, where Λ := I − (I + A)−1 ∈ L(X), f e := f ◦(I − Λ) : X → X
is locally Lipschitz continuous, and X := X 0 is defined as the extrapolation space
of X 1 generated by the realization of A in X 1 (see [1]).
In the extrapolation space X the resemblance of (the counterpart (1.11)) of (1.1)
with the classical strongly damped wave equation seems worthy to be mentioned
and exploited. To exhibit these types of properties it is then reasonable to substitute
w := vt , which leads to the first order problem
d v
[ ] + Aµ [ wv ] = F([ wv ]), t > 0, [ wv ]|t=0 = [ wv00 ]
dt w
in variables (t, v, w) with Aµ ∈ L(X × X) given in the matrix form as
0
−I
(1.14)
Aµ := Λ0 −I
µΛ = I−(I+A)−1 µ(I−(I+A)−1 )
(1.13)
and a locally Lipschitz map F : X × X → X × X being defined by
(1.15)
F([ wv ]) = f e0(v) , [ wv ] ∈ X × X.
In the above setting the linear operator Aµ defined in the product (extrapolation)
space X × X essentially resembles the strongly damped wave operator appearing
in the context of the classical wave equations (see e.g. [5, 8, 18]). The implicit
conclusion drawn from the above generalities is that the problem (1.1) should be
well posed in the Banach spaces W01,p (Ω) × W01,p (Ω), p ∈ (1, ∞), and that its
solutions should exhibit some smoothing properties at least at infinity and at least
in the (subcritical) case (SC). Also the existence of an attractor should follow within
the semigroup approach as for the problems enjoying a gradient-like structure.
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2570
A. N. CARVALHO AND J. W. CHOLEWA
To be more specific about our results, let A be the negative Dirichlet Laplacian
∆ in Lp (Ω), p ∈ (1, ∞), with the domain W 2,p (Ω) ∩ W01,p (Ω) and consider its closed
extension1 to X = (W01,p (Ω)) =: W −1,p (Ω).
As for the local well posedness of (1.1)-(1.2) we then obtain the following theorem.
Theorem 1.1. For each u0 , v0 ∈ W01,p (Ω), there exists τ (u0 , v0 ) > 0 and a unique
function u(·) ∈ C 2 ([0, τ (u0 , v0 )), W01,p (Ω) × W01,p (Ω)) satisfying
u(0) = u0 , ut (0) = v0 ,
and
utt (t) + µAut (t) + Au(t) + Autt (t) = f (u(t)) for each t ∈ [0, τ (u0 , v0 ))
provided that one of the conditions (H1), (H2) or (H3) in (1.7) holds.
Furthermore, functions u(t, u0 , v0 ) and ut (t, u0 , v0 ) depend continuously on its
variables (t, u0 , v0 ) ∈ [0, τ (u0 , v0 )) × W01,p (Ω) × W01,p (Ω) in the norm of W01,p (Ω) ×
W01,p (Ω) and either τu0 ,v0 = ∞ or u(t, u0 , v0 )W 1,p (Ω) +ut (t, u0 , v0 )W 1,p (Ω) → ∞
0
0
as t → τu−0 ,v0 .
As for the global well posedness result in H01 (Ω)×H01 (Ω), the nonlinear semigroup
of global solutions associated to the problem (1.1)-(1.2) is shown to possess much
of the properties of gradient systems (see [10]); namely,
Theorem 1.2. Assume that (1.4) holds, and that one of the conditions (H1), (H2)
or (H3) in (1.7) is satisfied with p = 2.
Then
i) for each u0 , v0 ∈ H01 (Ω) the solution u(t, u0 , v0 ) of (1.1) exists globally in
time,
0 ,v0 )
ii) relation S(t)([ uv00 ]) = uu(t,u
, t ≥ 0, u0 , v0 ∈ H01 (Ω), defines a C 0
(t,u
,v
)
t
0 0
semigroup on H01 (Ω) × H01 (Ω) with bounded orbits of bounded sets,
iii) the set E consisting of all the equilibria of {S(t)} has the form E = {(φ, 0),
φ ∈ E1 }, where E1 = {φ ∈ H01 (Ω) : Aφ = f (φ)} is a bounded subset of
C 2 (Ω̄),
iv) there is a continuous functional V : H01 (Ω) × H01 (Ω) → R such that
a) V(S(·)([ uv00 ])) is bounded from below and nonincreasing in [0, ∞) for
any [ uv00 ] ∈ H01 (Ω) × H01 (Ω),
b) [ uv00 ] ∈ H01 (Ω) × H01 (Ω), and V(S(·)([ uv00 ])) = const. in [0, ∞) implies
that [ uv00 ] ∈ E,
v) for each t ≥ 0 and [ uv00 ] ∈ W01,p (Ω) × W01,p (Ω) the following formula holds:
S(t)([ uv00 ]) = e−Aµ t [ uv00 ] + U (t, [ uv00 ]),
t
u0
U (t, [ v0 ]) :=
e−Aµ (t−s) Φ1 F(Φ−1 S(s)([ uv00 ]))ds,
(1.16)
0
where
e−Aµ t =
∞
(−1)k
k=0
k!
tk Akµ
1 To simplify the notation we will next use for the extensions (resp. restrictions) of the operator
A the same symbol A.
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SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2571
is the exponential function of the linear bounded operator −Aµ given by
(1.14) in W01,p (Ω) × W01,p (Ω) and Φ−1 : W01,p (Ω) × W01,p (Ω) → W0−1,p (Ω) ×
W0−1,p (Ω) is given by Φ−1 =
(I+A)
0
0
(I+A)
and Φ1 = (Φ−1 )−1 .
As for the asymptotic properties of the solutions of (1.1)-(1.2) the following
result is given.
Theorem 1.3. If (1.4) and (1.8) or (1.9) hold, then the semigroup {S(t)} associated to the evolution problem (1.1)-(1.2) in H01 (Ω) × H01 (Ω) has a global attractor
A given by (1.10).
When (1.8) holds, the global attractor A is bounded in W 2,q (Ω) ∩ W01,q (Ω) ×
2,q
W (Ω) ∩ W01,q (Ω) for each q ∈ (1, ∞) and thus compact in the space of Hölder
functions C 1+ν (Ω) × C 1+ν (Ω) for every ν ∈ (0, 1).
Theorems 1.1 and 1.2 are proved in Section 2. In Section 3 the asymptotic regularity properties of {S(t)} are obtained and the proof of Theorem 1.3 is completed
under the assumption (1.8). In Section 4 we give the proof of (1.6) under the assumption (1.9) using Alekseev’s nonlinear variation of constants formula (see [2],
[5]).
An open question remains whether the global attractor regularizes in the critical
case (1.9), that is, when the critical exponent ρ∗ is involved.
2. Local and global well posedness results
In what follows we will essentially use the properties of the double sided fractional
power scale {E α , α ∈ R} generated by (E, Ã), where E is a reflexive Banach space
and à with the domain E 1 is a sectorial positive operator in E. In particular, we
will use the properties of the scale corresponding to à = −∆+I in E = Lp (Ω) =: Ep
with the domain Ep1 = W 2,p (Ω) ∩ W01,p (Ω), where p ∈ (1, ∞) and ∆ denotes the
Dirichlet Laplacian.
1
−1
We recall that the extrapolation space Ep 2 of Ep2 is the completion of
1
12
Ep , (Ã 21 )−1 · 12 . We know that Ep2 coincides with W01,p (Ω) and from [1, PropoEp
−1
sition V.1.4.3] we know that the (adjoint) space ((Ep ) 2 ) coincides with Ep 2 and
1
with the completion of Ep , Ã− 2 · Ep .
Following [1, Proposition V.1.4.3] we remark that we identify the elements φ ∈
1
−1
1
Ep 2 with the functionals from (Ep2 ) = (W01,p (Ω)) in such a way that the norm
1
1
Ã− 2 φEp =: φ − 21 is equal to the norm φ
= φ(W 1,p (Ω)) =
2
Ep
L(Ep ,C)
φW −1,p (Ω) whenever φ ∈ Ep . For φ ∈
−1
Ep 2
0
\ Ep we then have φ
1
L(Ep2 ,C)
=
φ(W 1,p (Ω)) = φW −1,p (Ω) = limj→∞ Ã− 2 φj Ep for any suitable (Cauchy) se0
quence {φj } ⊂ E.
As for the representation of the functionals we obtain the following relation.
1
Lemma 2.1. If E2 = L2 (Ω) and à is the negative Dirichlet Laplacian plus any
multiple of the identity such that à is positive with the domain E21 = H 2 (Ω)∩H01 (Ω),
then
1
1
− 12
2
(2.1)
à φ, à ψE2 =
φψds, φ ∈ E 2N , ψ ∈ E22 .
Ω
N +2
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2572
A. N. CARVALHO AND J. W. CHOLEWA
E
1
2N
N +2
Proof. If E22 φn → φ ∈ E
Ã− 2 φ, Ã 2 ψE2
1
1
E
−1
2
1
2
, then φn →
φ and for ψ ∈ E22 we get
1
1
← Ã− 2 φn , Ã 2 ψE2 =
φn ψdx →
φψds,
2N
N +2
Ω
Ω
which proves the result.
−1
− 12
As for the properties of Aµ defined by (1.14) in Ep 2 ×Ep
lemma.
we prove the following
Lemma 2.2. Let p ∈ (1, ∞) and µ > 0. Suppose that A is the negative Dirichlet
Laplacian in Lp (Ω) with the domain W 2,p (Ω) ∩ W01,p (Ω) and consider its closed
extension to (W01,p (Ω)) =: W −1,p (Ω). Then Aµ is a bounded linear operator in
∞ (−1)k k k
−1
−1
Ep 2 × Ep 2 and the uniformly continuous semigroup e−Aµ t =
k=0 k! t Aµ
− 12
generated by −Aµ in Ep
− 12
× Ep
e−Aµ t (2.2)
satisfies the estimate
−1
2
L(Ep
−1
2
×Ep
)
≤ Mp e−ωt , t ≥ 0,
with certain positive constants Mp , ω.
1
Proof. If we write the equation (λI + Aµ ) φφ12 = ψ
ψ2 we will observe that
((λ2 +µλ+1)I−(µλ+1)(I+A)−1 )−1 λψ1 +µ(I−(I+A)−1 )ψ1 +ψ2
φ1
−1
φ2 =
λ((λ2 +µλ+1)I−(µλ+1)(I+A)−1 )
λψ1 +µ(I−(I+A)−1 )ψ1 +ψ2 −ψ1
−1
is well dewhenever the inverse operator (λ2 + µλ + 1)I − (µλ + 1)(I + A)−1
fined. From the spectral mapping theorem (see [7, p. 243]) we know that
σ((I + A)−1 ) − {0} = {(1 + ν)−1 , ν ∈ σ(A)},
which shows that σ((I + A)−1 ) ⊂ [0, (1 + λ1 )−1 ].
Since, for λ = −µ−1 ,
Im(λ2 (1 + µλ)−1 + 1) =
µImλ
1 2
1
2
)
+
(Reλ
+
−
(Imλ)
(1 + µReλ)2 + (µImλ)2
µ
µ2
we have that:
• the points in σ(−Aµ ) with Imλ = 0 must be of the form λ = − µ1 + µ1 eiθ .
For λ = − µ1 + µ1 eiθ we have that λ2 (1 + µλ)−1 + 1 = 2µ−2 (cos θ − 1) + 1.
Now, for µ2 > 4λ−1
θ ∈ R or for µ2 ≤ 4λ−1
1 (1+ λ1 ) and 1 (1 + λ1 ) and
0 < |θ| < θ0 := arccos
2+(2−µ2 )λ1
2+2λ1
, we have that λ = −µ−1 + µ−1 eiθ ∈
µλ1
, Imλ = 0;
ρ(−Aµ ). This shows that λ ∈ ρ(−Aµ ) whenever Reλ > − 2+2λ
1
2
• the points with Imλ = 0 which satisfy the condition λ (µλ + 1)−1 + 1 >
(1 + λ1 )−1 are in ρ(−Aµ ). Hence, we clearly have
ρ(−Aµ ) ∩ R ⊃ (−µ−1 , ∞).
Thus, ρ(−Aµ ) ⊃ λ ∈ C : Reλ > − min{µ−1 , µλ1 (2 + 2λ1 )−1 } and (2.2) is proved.
As for the isometric properties of the operator à we specifically mention the
following result (see [1, Corollary 1.3.9]).
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SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
−1
− 12
s− 12
Lemma 2.3. If p ∈ (1, ∞), s ∈ [0, ∞) and Φs : Ep 2 × Ep
given in the matrix form as
−s
0
,
(2.3)
Φs = (I+A)
0
(I+A)−s
→ Ep
2573
s− 12
× Ep
is
then Φs is an isometric isomorphism2 from Epr × Epr onto Epr+s × Epr+s for each
− 12
r ≥ − 12 ; in particular Φ1 is an isometric isomorphism from Ep
1
2
− 12
× Ep
onto
1
2
Ep × Ep .
2.1. Local well posedness result and the proof of Theorem 1.1. We first
prove the auxiliary lemma below.
Lemma 2.4. Let p ∈ (1, ∞), f : R → R, and A be as in Lemma 2.2. Then
f e (φ)(x) = f ((I + A)−1 φ(x)), x ∈ Ω,
defines an operator from W −1,p (Ω) into W −1,p (Ω) which is Lipschitz continuous in
bounded sets provided that one of the conditions (H1), (H2) or (H3) in (1.7) holds.
− 12
and choose arbitrary φ1 , φ2 ∈ B.
Proof. Let B be bounded in Ep
1
If condition (H1) is satisfied, then (I + A)−1 B is a bounded subset of Ep2 and
hence of C 0 (Ω). Since f is Lipschitz continuous on bounded subsets of R1 we have
f e (φ1 ) − f e (φ2 )
−1
2
Ep
≤ Cf e (φ1 ) − f e (φ2 )Ep0
≤ CL(I + A)−1 (φ1 − φ2 )Ep0 ≤ C̃Lφ1 − φ2 −1
2
Ep
.
When either (H2) or (H3) holds we use the Sobolev and Hölder inequalities to
get
f e (φ1 ) − f e (φ2 )
−1
Ep 2
−1
A) φ1 )
≤ Cf e (φ1 ) − f e (φ2 )E
Np
N +p
− f ((I + A)−1 φ2 )E N p
N +p
ρ−1
−1
−1
−1
≤ Ĉ(I +A) (φ1 −φ2 ) 12 1+(I +A) φ1 ρ−1
+(I
+
A)
φ
2 E N (ρ−1)
E N (ρ−1)
Ep
2
2
ρ−1
−1
+
(I
+
A)
φ
≤ C̃(I + A)−1 (φ1 − φ2 ) 12 1 + (I + A)−1 φ1 ρ−1
1
1
2
Ep
Ep2
Ep2
≤ C̃φ1 − φ2 − 21 1 + φ1 ρ−1
+ φ2 ρ−1
−1
−1
= Cf ((I +
Ep
≤ C̄φ1 − φ2 −1
2
Ep
Ep
2
Ep
2
.
The local well posedness result for (1.13) in W −1,p (Ω) × W −1,p (Ω) now follows.
Theorem 2.1. If the assumptions of Lemma 2.4 are satisfied and one of the conditions (H1), (H2) or (H3) in (1.7) holds, then for each v0 , w0 ∈ W −1,p (Ω) the problem
(1.13) has a unique solution [ wv ] (·, v0 , w0 ) ∈ C([0, τv0 ,w0 ), W −1,p (Ω) × W −1,p (Ω))
defined on the maximal interval of existence [0, τv0 ,w0 ) and continuously depending
2 Recall
that we consider the double sided fractional power scale {E α , α ∈ R} generated by
1
(E, Ã) with à = I + A; in particular for φ ∈ Ep2 we have that φ
φ
1
1
−
Ep 2
= (I + A)− 2 φEp .
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1
1
Ep2
= (I + A) 2 φEp and
2574
A. N. CARVALHO AND J. W. CHOLEWA
on the initial data, where either τv0 ,w0 = ∞ or v(t, v0 , w0 )X + w(t, v0 , w0 )X →
∞ as t → τv−0 ,w0 . The solution satisfies in W −1,p (Ω) the variation of constants
formula
t
0
−Aµ t v0
v
(2.4) [ w ] (·, v0 , w0 ) = e
ds, t ∈ [0, τv0 ,w0 ),
[ w0 ] +
e−Aµ (t−s) f e (v(s,v
0 ,w0 ))
0
with {e−Aµ t } defined in Lemma 2.2 and, in addition,
[ wv ] (·, v0 , w0 ) ∈ C 1 ([0, τv0 ,w0 ), W −1,p (Ω) × W −1,p (Ω)),
(2.5)
[ wvtt ] (·, v0 , w0 ) ∈ C 1 ([0, τv0 ,w0 ), W −1,p (Ω) × W −1,p (Ω)).
In particular, [ wv ] (·, v0 , w0 ) is a classical solution of (1.13) in X ×X := W −1,p (Ω)×
W −1,p (Ω).
−1
−1,p
2
Proof. Let p ∈ (1, ∞) and X =
(Ω). From Lemma 2.4, if B is a
Ep = W
φ1
φ2
bounded subset of X × X and ψ1 , ψ2 ∈ B, we obtain
φ1
φ1
φ2
φ2 e
e
1 ≤ c
−
F
=
f
(φ
)
−
f
(φ
)
.
F
1
2
−
ψ1
ψ2
ψ1 − ψ2 2
Ep
X×X
X×X
Hence, F takes X × X into X × X and is Lipschitz continuous on bounded sets
so that the semigroup theory applies (see [12, 15]), from which the results follow
easily.
The proof of Theorem 1.1. The local well posedness of (1.1) in W01,p (Ω) ×
W01,p (Ω) follows from Theorem 2.1 and Lemma 2.3. For this we define τ (u0 , v0 ) :=
τ(I+A)u0 ,(I+A)v0 and
u(t) = (I + A)−1 v(t), ut (t) = (I + A)−1 vt (t),
(2.6)
where [ vvt ] is the (uniquely defined on [0, τ (u0 , v0 )) solution of (1.13) from Theorem 2.1.
2.2. Global solutions to (1.1) and the proof of Theorem 1.2. Assuming
(1.4) we obtain below the global well posedness result for (1.1)-(1.2) and prove
Theorem 1.2.
Lemma 2.5. Under the assumptions and notation of Theorem 1.1 condition (1.4)
implies the existence of a constant C > 0 such that if u0 , v0 ∈ W01,p (Ω) and p ∈
[2, ∞), then the functions u(t, u0 , v0 ) and ut (t, u0 , v0 ) fulfill the estimate
2
, t ∈ [0, τ (u0 , v0 )).
(2.7)
ut 2 1 + u2 1 ≤ C 1 + u0 ρ+1
1 + v0 1
E22
E22
E22
E22
−1
Proof. From Theorem 2.1 function v = (I + A)u solves (1.11) in E2 2 . We thus
− 12
take the E2
product3 ·, ·
−1
2
E2
of each side of (1.11) with vt to get
1
d 1
vt 2 − 1 + v2 − 1 − (I + A)− 2 v2 − 1 − f e (v), vt − 12
E2
dt 2
E2 2
E2 2
E2 2
− 12
= −µ vt − 12 − (I + A) vt − 12 .
(2.8)
E2
3 Since
{E α , α ∈ R} is generated by (E, Ã) with à = I + A, then the product φ, ψ
−1
2
φ, ψ ∈ E2
in E2 .
E2
1
1
1
−
E2 2
of
is given by (I + A)− 2 φ, (I + A)− 2 ψE2 , ·, ·E2 being the (original) scalar product
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SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2575
The Poincaré inequality reads
1
1
(I + A) 2 φ2E2 ≥ (1 + λ1 )φ2E2 , φ ∈ E22 ,
(2.9)
which for φ = (I + A)−1 ψ translates into the estimates
−1
ψ2 − 1 ≥ (1 + λ1 )(I + A)− 2 ψ2 − 1 , ψ ∈ E2 2 ,
1
(2.10)
E2
2
E2
2
and implies
λ1
1
v2 − 1 ≤ v2 − 1 − (I + A)− 2 v2 − 1 ,
1 + λ1
E2 2
E2 2
E2 2
1
λ1
vt 2 − 1 ≤ vt 2 − 1 − (I + A)− 2 vt 2 − 1 .
1 + λ1
E2 2
E2 2
E2 2
(2.11)
From Lemma 2.1 we get
(I + A)− 2 f e (v), (I + A) 2 (I + A)−1 vt E2 =
1
1
f e (v)(I + A)−1 vt dx,
Ω
and hence
f e (v), vt = (I + A)− 2 f e (v), (I + A)− 2 vt E2
=
f ((I + A)−1 v)(I + A)−1 vt dx.
1
−1
E2 2
1
Ω
If F is the primitive function of f in R1 we then have
d
−1
−1
f ((I + A) v)(I + A) vt dx =
F ((I + A)−1 v)dx.
(2.12)
dt Ω
Ω
We now remark that (1.4) implies the existence of constants , C > 0, for which
t
1
F (t) =
f (s)ds ≤ (λ1 − )t2 + C, t ∈ R1 .
2
0
As a consequence we infer
F ((I + A)−1 v), 1E2 ≤
1
1
(λ1 − )(I + A)− 2 v)2 − 1 + C|Ω|,
2
E2 2
which with the aid of (2.10) reads
(2.13)
F ((I + A)−1 v)dx ≤
Ω
λ1 − v2 − 1 + C|Ω|.
2(1 + λ1 )
E2 2
Connecting (2.8)-(2.13) we get for
(2.14)
1
1
1
1
L(v, vt ) := vt 2 − 1 + v2 − 1 − (I + A)− 2 v2 − 1 −
2
2
2
2
2
E2
E2
E2 2
that
F ((I + A)−1 v)dx
Ω
d
µλ1
− 12
(L(v, vt )) = −µ vt − 12 − (I + A) vt − 12 ≤ −
vt 2 − 1 ≤ 0
E2
E2
dt
1 + λ1
E2 2
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2576
A. N. CARVALHO AND J. W. CHOLEWA
and hence
(2.15)
1
vt 2 − 1 +
v2 − 1 − C|Ω| ≤ L(v, vt ) ≤ L(v(0), vt (0)),
2
2(1 + λ1 )
E2 2
E2 2
as long as the solution v exists.
In the variables (t, u) we then have
(2.16)
1
2
2
2
2
2
2
u 1 ≤ v0 1 +(I +A) u0 E2 −u0 E2 −2
F (u0 )dx+2C|Ω|,
ut 1 +
1 + λ1
E22
E22
E22
Ω
where
F (u0 )L1 (Ω) ≤ const. 1 + u0 ρ+1
,
1
(2.17)
E22
since (1.7) implies that |F (s)| ≤ const.(1 + |s|ρ+1 ) for s ∈ R. Thus (2.7) follows
from (2.16) and (2.17) via the Poincaré inequality (2.9).
Proof of Theorem 1.2. Parts i) and ii) follow from Lemma 2.5 and Theorem 1.1.
To prove the boundedness of E1 in H01 (Ω) we first infer from Lemma 2.1 that
1
− 12
2
A f (u), A uE2 =
f (u)udx.
(2.18)
Ω
1
2
Since u ∈ E1 implies that u ∈ E2 and A u = A− 2 f (u), then
1
1
2
2
f (u)udx, u ∈ E1 .
A u, A uE2 =
(2.19)
1
2
1
Ω
With the aid of (1.4) and the Poincaré inequality we then get
(2.20)
2
2
A 2 u2E2 ≤ (λ1 − )u2E2 + C|Ω| ≤ (1 − λ−1
1 )A uE2 + C|Ω|
1
1
for some , C > 0 and hence
(2.21)
sup A 2 u2E2 ≤ C|Ω|λ1 −1 =: M1 .
1
u∈E1
Referring next to [3, Theorem 2.6], we infer that E1 is bounded in L∞ (Ω) and
1,q
2,q
∞). In particular, for any equilibrium
hence in W
φ (Ω) ∩ W0 (Ω) for each q ∈ (1,
1+ν
(Ω̄) for each ν ∈ (0, 1). Therefore
solution 0 the map f (φ) is of class C
φ ∈ C 2+ν (Ω̄) and E1 is bounded in C 2+ν (Ω̄) as a consequence of the Schauder type
estimates (see [9, Theorems 9.19, 6.6]).
For the proof of part iv) we come back to (2.14), defining
(2.22)
1
1
1
F (u)dx.
V(S(t)([ uv00 ])) := L(Φ−1 S(t)([ uv00 ])) = ut 2 1 + u2 1 − u2E2 −
2
2
2
E22
E22
Ω
1
Following the assumptions on f for φ, φ̃ ∈ E22 we have
F (φ) − F (φ̃)L1 (Ω) ≤ c̃|φ − φ̃|(1 + |φ|ρ + |φ̃|ρ )L1 (Ω)
≤ ĉφ − φ̃ 12 1 + φρ 1 + φ̃ρ 1 .
E2
E22
Properties a) and b) follow then from Lemma 2.5.
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E22
SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2577
For the proof of v) we combine (1.15), (2.4) and (2.6) to get
t
(I+A)u
(I+A)u(s)
−Aµ t (I+A)u0
(2.23)
e−Aµ (t−s) F (I+A)u
ds,
(I+A)ut (t) = e
(I+A)v0 +
(s)
t
0
1
t ≥ 0, [ uv00 ] ∈ E22 .
−1
−1
We also know that Φ1 ∈ L(E2 2 × E2 2 ) and Aµ commutes with Φ1 . Therefore,
e−Aµ t commutes with Φ1 and an application of Φ1 to both sides of (2.23) gives the
result.
3. Asymptotic regularity properties of {S(t)} and regularity
of the global attractor in the subcritical case
In this section we prove the asymptotic regularity properties of {S(t)} assuming
condition (1.8). We then obtain the existence and regularity properties of the
global attractor proving Theorem 1.3 under assumption (1.8). The existence of an
attractor for the case (1.9) will be proved in Section 4.
Before we proceed with the proof of Theorem 1.3 we need the following two
lemmas.
Lemma 3.1. If condition (1.8) holds, then f˜e = f e ◦(I + A) takes sets bounded in
σ̃−1
1
E22 into sets bounded in E
2N
N +2(1−σ̃)
and thus also in E2 2
with σ̃ = min{1, N2+2 −
1
ρ N 2−2 }. In particular, if N = 1 or N = 2, then f˜e takes bounded subsets of E22
into bounded subsets of Eq for every q ∈ (1, ∞).
1−σ
Proof. Let N ≥ 3 and recall that E2 2 → E
σ−1
2
1−σ
2
) . If 1 < ρ <
= (E2
E2
(3.1)
f ((I +A)φ)
e
σ̃−1
2
E2
N +2
N −2 ,
≤ c̃f (φ)E
2N
N −2(1−σ)
for σ ∈ [0, 1]. Recall also that
1
then (1.7) leads for φ ∈ E22 to the condition
≤ c̄1+|φ| E
ρ
2N
N +2(1−σ̃)
2N
N +2(1−σ̃)
ρ
≤ ĉ 1 + φ 1 .
E22
If N = 1 or N = 2, then
(3.2)
1
f (φ)Eq ≤ c̄1 + |φ|ρ Eq ≤ c̄ 1Eq + φρEρq ≤ ĉ 1 + φρ 1 , φ ∈ E22 ,
E22
1
2
since in this case E2 → Er with any r ∈ (1, ∞).
Lemma 3.2. Suppose that p ∈ (1, ∞), that A is as in Lemma 2.2 and that (1.4)
1
1
and (1.8) hold. Then, for any set B bounded in E22 × E22 and Mp , ω, ĉ as in (2.2),
(3.1) we have
(α) in the case N ≥ 3,
t
1+σ̃
sup sup
e−Aµ (t−s) Φ1 F(Φ−1 S(s)([ uv00 ]))ds 1+σ̃
u0 ∈B t≥0
E2 2 ×E2 2
0
[ v0 ]
M2 ĉ ρ
≤
u0 1
1+
sup
ω
E22
[ u0 ]∈γ + (B)
v0
with σ̃ := min{1,
N
2
+ 1 − ρ( N2 − 1)}
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2578
A. N. CARVALHO AND J. W. CHOLEWA
and
(β) in the case N = 1 or N = 2,
t
sup sup
e−Aµ (t−s) Φ1 F(Φ−1 S(s)([ uv00 ]))ds 1 1
u0 ∈B t≥0
Ep ×Ep
0
[ v0 ]
Mp ĉ ρ
≤
u0 1 .
1+
sup
ω
E22
[ u0 ]∈γ + (B)
v0
Proof. Recalling (1.16) and Lemma 2.3 we have
(3.3)
U (t, [ uv00
])
1+σ̃
2
E2
1+σ̃
2
×E2
t
= Φ−1− σ̃ Φ1
e−Aµ (t−s) F(Φ−1 S(s)([ uv00 ]))ds
2
t
= Φ− σ̃
e−Aµ (t−s) F(Φ−1 S(s)([ uv00 ]))ds
2
− 12
We remark that Φ σ̃ ∈ L(E2
2
U (t, [ uv00 ])
1+σ̃
E2 2
1+σ̃
×E2 2
t
−Aµ (t−s) ≤
e
0
.
−1
2
σ̃−1
σ̃−1
and from (3.3)
t
=
e−Aµ (t−s) Φ− σ̃ F(Φ−1 S(s)([ uv00 ]))ds
2
0
−1
L(E2 2
−1
2
×E2
−1
2
×E2
× E2 2 ) and Φ σ̃ commutes with Aµ and hence with
e−Aµ t . Thus Φ−1
= Φ− σ̃ commutes with e−Aµ t on E2 2 × E2 2
σ̃
2
2
we get
(3.4)
−1
2
E2
0
−1
2
E2
0
−1
×E2 2
)
Φ− σ̃2 F(Φ−1 S(s)([ uv00 ]))
−1
2
E2
−1
2
E2
−1
2
×E2
−1
2
×E2
ds.
1
1
Since via Theorem 1.2 ii) the positive orbit γ + (B) is bounded in E22 × E22 , Lemma
3.1 implies
Φ σ̃ F(Φ−1 S(s)([ uv 0 ])) − 1 − 1 = F(Φ−1 S(s)([ uv 0 ])) σ̃−1
σ̃−1
0
0
−2
E2 2 ×E2 2
E2 2 ×E2 2
(3.5)
ρ
≤ ĉ 1 +
sup
u0 1 , [ uv00 ] ∈ B, s ≥ 0.
E22
[ uv00 ]∈γ + (B)
With (3.5) and with the exponential estimate of Lemma 2.2, part (α) now follows
easily.
For the proof of part (β) we take σ̃ = 1, which is possible thanks to the estimate
(3.2). We then replace symbol E2 with Ep in (3.3)-(3.5) to get
t
e−Aµ (t−s) Φ− 12 F(Φ−1 S(s)([ uv00 ]))ds − 12 − 21
U (t, [ uv00 ])Ep1 ×Ep1 = Ep ×Ep
0
(3.6)
t
−Aµ (t−s) ≤
e
− 21 − 21 Φ− 12 F(Φ−1 S(s)([ uv00 ])) − 21 − 21 ds,
L(Ep
0
×Ep
×Ep
Ep
)
where, thanks to (3.2), for
] ∈ B and s ≥ 0 we have
u
−1
0
Φ− 1 F(Φ S(s)([ v ])) − 1 − 1 = F(Φ−1 S(s)([ uv 0 ]))
[ uv00
2
1
0
Ep
2
×Ep
2
≤ ĉ 1 +
1
0
Ep ×Ep
sup
u0 ρ 1 .
E22
[ uv00 ]∈γ + (B)
The result now follows via (2.2).
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SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2579
Corollary 3.1. Under the assumptions of Lemma 3.2 the semigroup S(t) : H01 (Ω)×
H01 (Ω) → H01 (Ω) × H01 (Ω), t ≥ 0, is asymptotically compact (equivalently asymptotically smooth).
1+σ̃
1
Proof. Since the embedding E2 2 → E12 is compact, thanks to (1.16) and Lemmas 2.2, 3.2 the semigroup {S(t)} is the sum of an exponentially decaying linear
semigroup and of a family of maps U (t) compact for t > 0. The result thus follows
from [10, Lemma 3.2.3].
Remark 3.1. Theorem 1.2 and Corollary 3.1 ensure the existence of a global attractor A in the presence of the assumption (1.8). Indeed, {S(t)} is the asymptotically
smooth semigroup with bounded orbits of bounded sets whereas the set of equilibria
is bounded and attracts points. The result thus follows from [10, Theorem 3.4.6].
A bootstrapping procedure for the solutions in the attractor will lead to the
regularity result of Theorem 1.3. The next four lemmas contain a proof of this fact.
Lemma 3.3. Suppose that the assumptions of Lemma 3.2 hold and let A denote
a global attractor for {S(t)} in H01 (Ω) × H01 (Ω) (see Remark 3.1). Then
(a) A is a bounded subset of H 1+σ̃ (Ω) ∩ H01 (Ω) × H 1+σ̃ (Ω) ∩ H01 (Ω) if N ≥ 3,
(b) A is bounded in W 2,q (Ω)∩W01,q (Ω)×W 2,q (Ω)∩W01,q (Ω) with any q ∈ (1, ∞)
if N ∈ {1, 2}.
Proof. Note that A is invariant and, in particular, any point from A can be viewed
as an H01 (Ω) × H01 (Ω)-limit of a sequence of the form
(3.7)
0n
0n
0n
0n
S(tn )([ uv0n
]) = e−Aµ tn [ uv0n
] + U (tn , [ uv0n
]), where tn ∞ and {[ uv0n
]} ⊂ A.
Since A is bounded in H01 (Ω) × H01 (Ω), conditions (a), (b) follow from Lemmas 2.2,
3.2 and from the weak convergence property.
Lemma 3.4. Under the assumptions of Lemma 3.3, if N ≥ 3, then A is bounded
in the norm of H 2 (Ω) ∩ H01 (Ω) × H 2 (Ω) ∩ H01 (Ω).
Proof. If σ̃ = 1, then the result coincides with Lemma 3.3 (a).
N +2
Suppose that σ̃ < 1 and set r0 := σ̃, r1 = min{1, N
−2 r0 }. Since 1 < ρ <
∗
∗
ρ
ρ , we infer from (1.7) that |f (s)| ≤ c̄(1 + |s| ) for s ∈ R and hence
e
(3.8) f ((I + A)φ)
r1 −1
E2 2
≤ c̃f (φ)E
2N
N +2(1−r1 )
N +2
N −2
=
1+r0
∗
≤ ĉ 1 + φρ 1+r0 , φ ∈ E2 2 .
E2
2
As a consequence of Theorem 1.3 and Lemma 3.3 the set A is invariant and bounded
1+r0
2
in E2
(3.9)
1+r0
2
× E2
, so that (3.8) implies
Φ− r1 F(Φ−1 S(s)([ uv00 ])) − 1 − 1 = F(Φ−1 S(s)([ uv00 ])) r1 −1
r1 −1
2
E2 2 ×E2 2
E2 2 ×E2 2
ρ∗
≤ ĉ 1 + sup u0 1+r0 , [ uv00 ] ∈ A, s ≥ 0.
E2 2
[ uv00 ]∈A
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2580
A. N. CARVALHO AND J. W. CHOLEWA
Estimating in a similar manner as in Lemma 3.2 we get
(3.10)
U (t, [ uv00 ])
1+r1
1+r1
E2 2 ×E2 2
t
−Aµ (t−s) ≤
e
0
−1
L(E2 2
t
= Φ−1− r21 Φ1 e−Aµ (t−s) F(Φ−1 S(s)([ uv00 ]))ds
−1
×E2 2
)
−1
2
E2
0
Φ− r1 F(Φ−1 S(s)([ uv00 ]))
2
−1
2
E2
−1
2
×E2
−1
2
×E2
ds
and hence (using (2.2) and (3.9))
sup sup U (t, [ uv00 ])
(3.11)
[ uv00 ]∈A t≥0
1+r1
2
E2
1+r1
2
×E2
< ∞.
Recalling that any point from A can be viewed as an H01 (Ω) × H01 (Ω)-limit of a
sequence of the form given in (3.7), via weak convergence we infer from (2.2), (3.7),
(3.11) that
sup [ uv00 ]
(3.12)
[ uv00 ]∈A
If r1 =
N +2
N −2 r0
1+r1
2
×E2
< ∞.
< 1, then we repeat the above procedure and obtain that
1+r2
2
(3.13)
1+r1
2
E2
A is bounded in E2
1+r2
2
× E2
N +2 2
, where r2 = min{1, ( N
−2 ) r0 }.
In a finite number of k-steps we will then reach rk = 1, which gives the result.
Lemma 3.5. Under the assumptions and notation of Lemma 3.3, if N ≥ 3 and
a number q ∈ (1, ∞) is such that A is bounded in Eq1 × Eq1 , then A is bounded in
Eq1∗ × Eq1∗ , where
q
N
in the case when N > 2q,
ρ∗ N − 2q
q ∗ is an arbitrary number from (1, ∞) when N ≤ 2q.
q∗ =
(3.14)
N +2
∗
Proof. For 1 < ρ < N
−2 = ρ we infer from (1.7) that
(3.15)
∗
∗
f e ((I + A)φ)Eq∗ ≤ c̃f (φ)Eq∗ ≤ c̄1 + |φ|ρ Eq∗ ≤ ĉ 1 + φρE 1 , φ ∈ Eq1 .
q
Since A is invariant and bounded in Eq1 ×Eq1 , (3.15) implies for [ uv00 ] ∈ A and s ≥ 0,
Φ− 1 F(Φ−1 S(s)([ uv 0 ])) − 1 − 1 = F(Φ−1 S(s)([ uv 0 ]))
0
0
Eq∗ ×Eq∗
2
Eq∗2 ×Eq∗2
∗
≤ ĉ 1 + sup u0 ρE 1 .
q
[ uv00 ]∈A
From this and (2.2) we thus have for a certain constant D > 0 and all [ uv00 ] ∈ A,
t ≥ 0,
t
e−Aµ (t−s) F(Φ−1 S(s)([ uv00 ]))ds − 12 − 12
U (t, [ uv00 ])E 1∗ ×E 1∗ = Φ−1− 12 Φ1
q
q
t
−Aµ (t−s) ≤
e
0
Eq∗ ×Eq∗
0
−1
−1
L(Eq∗2 ×Eq∗2 )
Φ− 12 F(Φ−1 S(s)([ uv00 ]))
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−1
−1
Eq∗2 ×Eq∗2
ds ≤ D.
SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2581
Now viewing points from A as H01 (Ω) × H01 (Ω)-limits of sequences of the form (3.7)
we obtain via (2.2), (3.7) and the weak convergence property that
sup [ uv00 ]E 1∗ ×E 1∗ < ∞.
[ uv00 ]∈A
q
q
The next simple lemma establishes, together with Lemma 3.5 and Lemma 3.4
the asymptotic bootstrapping procedure.
Lemma 3.6. If N ≥ 3 and ρ∗ =
N +2
N −2 ,
then the sequence {qk } defined as
qk
N
N
N
> qk , qk+1 = N if
≤ qk ,
if
ρ∗ N − 2qk
2
2
has the property that qk = N for all k sufficiently large.
(3.16)
q0 = 2,
qk+1 =
Proof. Note that q1 ≥ q0 and hence qk+1 ≥ qk for all k ∈ N. If qk < N2 for
∗
each k ∈ N, then we would have q0 ≤ lim qn = N (ρ2ρ∗−1) = 2 NN+2 < q0 , which is
impossible.
With these, all the conclusions of Theorem 1.3 connected with the subcritical
case (1.8) are proved.
4. Asymptotic smoothness of {S(t)} and the attractor
in the critical case
This section is devoted to the proof of existence of the attractor for the semigroup
{S(t)} associated in H01 (Ω) × H01 (Ω) to the evolution problem (1.1)-(1.2) in the
critical case (1.9), completing the proof of Theorem 1.3. It only remains to prove
the asymptotic compactness of {S(t)}. To that end we will employ Alekseev’s
Nonlinear Variation of Constants Formula (see [2, 5]). We first need to prove
differentiability of the map f e : H −1 (Ω) → H −1 (Ω).
2p
Lemma 4.1. Suppose that f ∈ C 1 (R, R), N > p, and |f (s)| ≤ c(1 + |s| N −p ),
s ∈ R, for some c > 0. Then the map f e = f ◦ (I + A)−1 is of the class
C 1 (W −1,p (Ω), W −1,p (Ω)).
Proof. Since (I + A)−1 is a linear isomorphism from W −1,p (Ω) into W 1,p (Ω), it is
Fréchet differentiable between these spaces with the derivative (I +A)−1 . Therefore,
it remains to ensure that f is continuously Fréchet differentiable from W 1,p (Ω) into
W −1,p (Ω).
Under our assumptions a sufficient condition for this is
(4.1)
f (θ(u + h) + (1 − θ)u) − f (u)
L
N
2
(Ω)
→ 0 when hW 1,p (Ω) → 0,
where θ varies in the interval [0, 1].
Since we know that |f (θ(u + hn ) + (1 − θ)u) − f (u)| tends to zero a.e. in Ω
as hn W 1,p (Ω) → 0, we will apply in what follows the Inverse Dominated Convergence Theorem. As a consequence of this theorem we infer from the fact that
→ 0 that there exists a subsequence {nk }, for which hnk → 0 a.e. in
hn N p
L N −p (Ω)
Ω and also
Np
Np
|hnk (x)| N −p ≤ |g(x)| N −p for a.e. x ∈ Ω,
Np
with g being a certain element of L N −p (Ω).
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2582
A. N. CARVALHO AND J. W. CHOLEWA
Fixing u and recalling the growth of f we have a.e. in Ω,
Np
Np
Np
Np
|f (θ(u+hnk )+(1−θ)u)−f (u)| 2 ≤ c(1+|hn | N −p +|u| N −p ) ≤ c(1+|g| N −p +|u| N −p ),
N
Np
where u, g ∈ L N −p (Ω). Condition (4.1) now follows easily.
Next we introduce a decomposition of the nonlinearity f which plays an essential
role in the proof of condition (1.6).
Lemma 4.2. Suppose that f ∈ C 1 (R, R) and lim sup|s|→∞ f (s)s−1 = µ ∈ R.
Then there are functions f1 ∈ C 1 (R, R) and f2 : R → R linear such that f (s) =
f1 (s) + f2 (s) and sf1 (s) ≤ 0 for all s ∈ R,
Proof. It is sufficient to define f2 (s)
= f (0) + Ks, where K ≥ max{µ, f (0)} is a
constant for which s−1 f (s) − f (0) < K for all s = 0.
A decomposition of the nonlinear term F and of the semigroup of global solutions
of (1.13) can now be established in accordance with the decomposition of f .
Lemma 4.3. Suppose that f ∈ C 1 (R, R), lim sup|s|→∞ f (s)s−1 = µ ∈ R, N > p,
2p
and |f (s)| ≤ c(1 + |s| N −p ), s ∈ R, for some c > 0.
Under these assumptions the following conditions hold.
(C1) There are functions f1 ∈ C 1 (R, R) and f2 : R → R linear such that f (s) =
f1 (s) + f2 (s), sf1 (s) ≤ 0 for s ∈ R, and
0
0
(4.2)
F([ wv ]) = f1 ((I+A)−1 v) + f2 ((I+A)−1 v) =: F1 ([ wv ]) + F2 ([ wv ])
for [ wv ] ∈ W −1,p (Ω) × W −1,p (Ω), where
F1 ∈ C 1 (W −1,p (Ω) × W −1,p (Ω), W −1,p (Ω) × W −1,p (Ω)),
(4.3)
F2 ∈ C 1 (W −1,p (Ω) × W −1,p (Ω), W 1,p (Ω) × W 1,p (Ω)).
(C2) If T̃ (t, [ wv00 ]) solves
(4.4)
d
dt
[ w̃ṽ ] + Aµ [ w̃ṽ ] = F1 ([ w̃ṽ ]), t > 0,
[ w̃ṽ ]t=0 = [ wv00 ]
and if T (t, [ wv00 ]) solves
(4.5)
d
dt
[ wv ] + Aµ [ wv ] = F1 ([ wv ]) + F2 ([ wv ]), t > 0,
then
(4.6)
T (t, [ wv00
]) =
T̃ (t, [ wv00
[ wv ]t=0 = [ wv00 ] ,
t
(∂ T̃ )(t − s, T (s) [ wv00 ])F2 (T (s, [ wv00 ]))ds,
]) +
t > 0,
0
where ∂ T̃ denotes the partial derivative of the map
(0, ∞) × W −1,p (Ω) × W −1,p (Ω) (t, [ wv00 ]) → T̃ (t, [ wv00 ]) ∈ W −1,p (Ω) × W −1,p (Ω)
with respect to [ wv00 ] and
0
F2 ([ wv ]) = K(I+A)−1 v+f (0) , [ wv ] ∈ W −1,p (Ω) × W −1,p (Ω).
(C3) Function (∂ T̃ )(·, [ wv00 ])F2 ( ŵv̂00 ) is a mild solution of
(4.7)
d
W + Aµ W = F1 (T̃ (t, [ wv00 ])W, t > 0,
dt
W(0) = F2 (
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v̂0 ŵ0 ).
SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2583
Proof. Condition (C1) follows from Lemmas 4.1 and 4.2.
Since Aµ can be viewed as a sectorial operator in W −1,p (Ω) × W −1,p (Ω) and
(4.3) holds, then (C2) follows from [5, Lemma 7]. For the proof of (C3) we refer to
[12, Theorem 3.4.4].
In Lemma 4.4 below we will obtain some suitable estimates for T̃ (·, ·), T (·, ·) and
W(·).
Lemma 4.4. If (1.4) and (1.9) are satisfied, then Lemma 4.3 applies with p = 2
and the following properties hold.
(P1) H −1 (Ω) × H −1 (Ω) [ wv00 ] → T̃ (t, [ wv00 ]) ∈ H −1 (Ω) × H −1 (Ω) (where t ≥ 0)
is a C 0 semigroup of global solutions to (4.4) with bounded (positive) orbits
of bounded sets.
(P2) H −1 (Ω) × H −1 (Ω) [ wv00 ] → T (t, [ wv00 ]) ∈ H −1 (Ω) × H −1 (Ω) (where t ≥ 0)
is a C 0 semigroup of global solutions to (4.5) with bounded (positive) orbits
of bounded sets.
−1
−1
σ−1
(P3) If B is a bounded subset of E2 2 ×E2 2 and [ χχ12 ] ∈ E2 2 for some σ ∈ [0, 1],
then there is a constant MB such that
(4.8)
χ1
σ−1 ≤ MB [ χ2 ] σ−1
σ−1 .
sup sup F1 (T̃ (s, [ wv00 ])) [ χχ12 ] σ−1
E2 2 ×E2 2
E2 2 ×E2 2
s≥0 [ v0 ]∈B
w0
−1
−1
(P4) If B is a bounded subset of E2 2 ×E2 2 , then for a certain constant NB > 0,
sup sup F2 (T (s, [ wv00 ]))E2 ×E2 ≤ NB .
s≥0 [ v0 ]∈B
w0
(4.9)
−1
−1
(P5) If τ > 0 and B is bounded in E2 2 × E2 2 , then there is a constant R(τ, B)
such that for each ŵv̂00 ∈ {T (s, [ wv00 ]), s ≥ 0, [ wv00 ] ∈ B} =: B̂ the mild
solution W of (4.7) fulfills the estimate
sup W(t)E2 ×E2 ≤ R(τ, B);
(4.10)
t∈[0,τ ]
(4.11)
in particular,
for all ŵv̂00 ∈ B̂ the corresponding values of W(t) until
− 12
−1
× E2 2 .
the time τ lie in a certain compact subset of E2
Proof. Properties (P1), (P2) follow from Theorem 2.1 and properties of the functional L in Lemma 2.5, which lemma is applicable since both f and f1 satisfy
(1.4).
0
For the proof of (P3) we first note that F1 ([ zz12 ]) [ χχ12 ] = f1 ((I+A)−1 z1 )(I+A)−1 χ1
and hence
F1 ([ zz12 ]) [ χχ12 ] σ−1
2
E2
σ−1
2
×E2
≤ df1 ((I + A)−1 z1 )(I + A)−1 χ1 E
2N
N +2(1−σ)
≤ df1 ((I + A)−1 z1 )E N (I + A)−1 χ1 E
2
≤
df1 ((I
+ A)
−1
z1 )E N (I + A)
−1
2
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σ−1
2
E2
1+σ
2
E2
2
= df1 ((I + A)−1 z1 )E N χ1 χ1 2N
N −2(1+σ)
.
2584
A. N. CARVALHO AND J. W. CHOLEWA
With the aid of (1.9) we next obtain
(4.12)
−1
σ−1 ≤ dc(1 + |(I + A)
F1 ([ zz12 ]) [ χχ12 ] σ−1
z1 | N −2 )E N χ1 σ−1
2
E2 2 ×E2 2
E2 2
4
4
≤ cd 1E N + (I + A)−1 z1 EN −2
≤ c̃ 1 + z1 N−−21 χ1 χ1 σ−1
2N
2
4
2
E2
N −2
E2
σ−1
2
.
E2
2
Property (P3) now follows from (P1) and (4.12).
In the proof of (P4) we note that f2 (s) = f (0) + Ks is a linear function and
therefore
F2 ([ χχ12 ])E2 ×E2 = f (0) + K(I + A)−1 χ1 E2 ≤ d˜ 1 + χ1 − 21 .
(4.13)
E
2
Thus (P4) follows from (P2) and (4.13).
To complete the proof we write the variation of constants formula for W:
t
v̂0 −Aµ t
(4.14)
W(t) = e
F2 ( ŵ0 ) +
e−Aµ (t−s) F1 (T̃ (s, [ wv00 ]))W(s)ds,
0
and apply Φ− 12 to get, with the aid of (P3), (P4) and (2.2),
(4.15)
W(t)E2 ×E2 = Φ− 12 W(t) − 21 − 21 ≤ e−Aµ t Φ− 12 F2 ( ŵv̂00 ) − 12 − 12
E2 ×E2
E2 ×E2
t
+
e−Aµ (t−s) − 12 − 21 Φ− 12 F1 (T̃ (s, [ wv00 ]))W(s) − 12 − 21 ds
E2 ×E2
E2 ×E2
0
t
M2 e−ω(t−s) F1 (T̃ (s, [ wv00 ]))W(s)E2 ×E2 ds
≤ M2 e−ωt NB +
0
t
M2 MB W(s)E2 ×E2 ds.
≤ M2 NB +
0
Relation (4.10) is now a consequence of Gronwall’s inequality and (4.11) follows via
−1
compact embedding E2 → E2 2 .
To conclude the proof of asymptotic compactness for {S(t)} it only remains to
prove that {T̃ (t)} decays exponentially, uniformly in bounded sets.
Lemma 4.5. Given r > 0 there are constants c(r) > 0 and β(r) > 0 such that
sup T̃ (t, [ wv00 ]) − 12 − 12 : [ wv00 ] − 12 − 12 ≤ r ≤ c(r)e−β(r)t .
E2
×E2
E2
×E2
Proof. The proof follows as in Proposition 3 of [5] using the functional
Lδ (v, vt ) = L(v, vt ) + δv, vt −1
2
E2
.
In fact, let
ṽ
ṽt
=: T̃ (t, [ wv00 ]),
t≥0
and
[ wv00 ]
−1
2
E2
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−1
2
×E2
≤ r.
SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER IN TIME
2585
Choosing δ > 0 suitably small we have that
µλ1
d
(Lδ (v, vt )) ≤ −
+ δ vt 2 − 1 + δv, −µΛvt − Λv + f1e (v) − 21
E2
dt
1 + λ1
E2 2
µλ1
≤−
+ δ vt 2 − 1 − δv, µΛvt − 21 − δv, Λv − 12
E2
E2
1 + λ1
E2 2
µλ1
δ
≤−
vt 2 − 1 − v − 12 .
E2
2 + 2λ1
2
E2 2
On the other hand, using (2.11) we obtain, for suitably small δ, that
Lδ (v, vt ) =
(4.16)
1
1
1
1
vt 2 − 1 + v2 − 1 − (I + A)− 2 v2 − 1 + δv, vt − 12
E2
2
2
2
E2 2
E2 2
E2 2
−
F1 ((I + A)−1 v)dx
Ω
∗
1
1
2
≤ vt 2 − 1 + v2 − 1 + δv, vt − 21 + ĉ(1 + vρ −−1
1 )v − 1
2
2
E
2
2
E2
E2
E2 2
E2 2
2
≤ c̄(r) vt 2 − 1 + v2 − 1 .
E2
2
E2
2
The proof of the result now follows easily.
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Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação,
Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São
Carlos SP, Brazil
E-mail address: [email protected]
Institute of Mathematics, Silesian University, 40-007 Katowice, Poland
E-mail address: [email protected]
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