Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 07, pp. 1–33.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
PULLBACK ATTRACTORS FOR A CLASS OF SEMILINEAR
NONCLASSICAL DIFFUSION EQUATIONS WITH DELAY
HAFIDHA HARRAGA, MUSTAPHA YEBDRI
Abstract. In this article, we analyze the existence of solutions for a nonclassical reaction-diffusion equation with critical nonlinearity, a time-dependent
force with exponential growth and delayed force term, where the delay term
can be entrained by a function under assumptions of measurability. Using a
priori estimates we obtain the pullback D-absorbing process and the pullback
ω-D-limit compactness that allow us to prove the existence of the pullback
D-attractors for the associated process to the problem.
1. Introduction and statement of the problem
The nonclassical diffusion equations occur as models in mechanics, soil mechanics and heat conduction theory (see for example [1, 2, 8, 9, 14]). In recent years,
the existence of pullback attractors has been proved for some nonclassical diffusion equations, see for example [16, 19, 21, 22, 23]. Functinal partial differential
equations is the subject of intensive studies.
For the functional partial differential equation
∂
∂
u(t, x) − ∆ u(t, x) − ∆u(t, x) = b(t, u(t − ρ(t))(x) + g(t, x)
∂t
∂t
u = 0 on (τ, ∞) × ∂Ω ,
u(τ + θ, x) = ϕ(θ, x),
in (τ, ∞) × Ω ,
(1.1)
τ ∈ R , θ ∈ [−r, 0], x ∈ Ω ,
without critical non-linearity, the long-time behavior, and especially the pullback
attractors has been studied in [7]. There the author studied the pullback asymptotic
behavior of solutions in the phase-spaces C([−r, 0]; H01 (Ω)) and C([−r, 0]; H01 (Ω) ∩
H 2 (Ω)).
In [16], the equation without delay
∂
∂
u(t, x) − ε∆ u(t, x) − ∆u(t, x) + f (u(t, x)) = g(t, x)
∂t
∂t
u = 0 on (τ, ∞) × ∂Ω
u(τ, x) = u0 (x),
in (τ, ∞) × Ω
τ ∈ R, x ∈ Ω ,
2010 Mathematics Subject Classification. 35R10, 35K57, 35B41.
Key words and phrases. Pullback attractors; nonclassical reaction-diffusion equations;
critical exponent, delay term.
c
2016
Texas State University.
Submitted June 4, 2015. Published January 6, 2016.
1
(1.2)
2
H. HARRAGA, M. YEBDRI
EJDE-2016/07
in the phase-space H01 (Ω) is treated. It is proved that the existence of a pullback attractor where the non-linearity f has a critical exponent in the interval
N +2
4
0, min{ N
−2 , 2 + N } with N ≥ 3. On unbounded domain, in [23], the existence
of pullback attractor to the solutions in H 1 (RN ) of the following equation without
delay
∂
∂
u(t, x) − ∆ u(t, x) − ∆u(t, x) + u(t, x) + f (u(t, x)) = g(t, x)
∂t
∂t
in (τ, ∞) × RN ,
u(τ, x) = uτ (x),
(1.3)
τ ∈ R, x ∈ RN ,
is treated, where the nonlinearity has a critical exponent p ≤ N 2−2 for N ≥ 3.
In this article, we consider the functional partial differential equation
∂
∂
u(t, x) − ∆ u(t, x) − ∆u(t, x) + f (u(t, x)) = b(t, ut )(x) + g(t, x)
∂t
∂t
with with the boundary and initial conditions
u=0
in (τ, ∞) × Ω ,
on (τ, ∞) × ∂Ω ,
u(τ, x) = u0 (x),
τ ∈ R, x ∈ Ω ,
(1.4)
u(τ + θ, x) = ϕ(θ, x), θ ∈ (−r, 0), x ∈ Ω ,
where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω. The
equation (1.4) without the term ∆ ∂u
∂t , is a classical equation with delay. Many
works have dealed with such equation, see for example [4, 5, 10, 11, 15, 17, 18].
It has been treated in different phase-spaces and the delay term is driven by a
function under measurability condition and the nonlinearity is given by different
assumptions. For more details on differential equations with delay we refer the
reader to [6] and [20].
It is well known that the compact Sobolev embedding can be applied to obtain
the existence of pullback D-attractor as well as the higher regularity of the solution
of the equation, e.g., although the initial conditions only belong to a weaker topological space, the solution will belong to a stronger topological space with higher
regularity. The equation (1.4) contains the term ∆ ∂u
∂t , this involves that the solution has no higher regularity and so the compact Sobolev embedding can not be
applied to obtain the existence of a pullback D-attractor. This is similar to the
hyperbolic case.
In this article, we prove the existence of a pullback D-attractor. It is well known
that for the existence of pullback D-attractors, the key point is to find a bounded
family of pullback D-absorbing sets then the pullback w-D-limit compactness for
the process corresponding to the solution of our problem. As noticed before, because of the term ∆ ∂u
∂t , the pullback w-D-limit compactness for the process can
not be proved by the compact Sobolev embedding. The nonlinearity with critical
exponent makes also some barriers. To overcome these difficulties, we apply the decomposition techniques and a method used in [17] to satisfy the pullback w-D-limit
compactness of the process with delay. It is based on the concept of the Kuratowski
measure of noncompactness of a bounded set as well as some new estimates of the
equicontinuity of the solutions.
This article is organized as follows. In section 2 useful results on nonautonomous
dynamical systems and pullback D-attractor theory are recalled. In section 3 deals
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DIFFUSION EQUATIONS WITH DELAY
3
with the main results; we will prove the existence of the solutions using the FaedoGalerkin approximations; also, the uniqueness and the continuous dependence of
the solutions with respect to the initial conditions are proved. Then we prove the
existence of the pullback D-attractor.
2. Preliminaries
At first, we give some notation which will be used throughout this paper. Let
Ω ⊂ RN (N ≥ 3) be a bounded domain with smooth boundary ∂Ω, the norm and
the inner product in L2 (Ω) are denoted by k·k and h·, ·i, respectively, and we denote
by k∇k and h∇·, ∇·i the norm and the inner product of H01 (Ω), respectively. The
norm in the Banach space Y will be denoted by k·kY . Let c be an arbitrary positive
constant, which may be different from line to line and even in the same line.
To study problem (1.4), we need some assumptions: The nonlinear function
f ∈ C 1 (R, R) satisfies
f (u)u ≥ −c1 u2 − c2 ,
(2.1)
0
(2.2)
f (u) ≥ −c3 , f (0) = 0,
α
|f (u)| ≤ k(1 + |u| ),
(2.3)
uf (u) − c4 F (u)
≥ 0,
u2
F (u)
lim inf 2 ≥ 0 ,
|u|→∞ u
lim inf
|u|→∞
(2.4)
(2.5)
N +2
4
where 0 < α < min{ N
−2 , 2+ N } (is called a critical exponent since the nonlinearity
f is not compact in this case i.e. for a bounded subset B ⊂ H01 (Ω), in general,
+4
f (B) is not precompact in Lq (Ω) where q = 2N
αN ), and c1 , c2 , c3 , c4 , k are positive
constants, λ1 > 0 is the first eigenvalue of −∆ in Ω with
R u the homogeneous Dirichlet
condition such that λ1 > max{c1 , c3 }, and F (u) = 0 f (s)ds. We infer from (2.4)
and (2.5) that for any δ > 0 there exist positive constants cδ , c0δ such that
uf (u) − c4 F (u) + δu2 + c0δ ≥ 0 ,
2
F (u) + δu +
c0δ
≥ 0,
∀u ∈ R ,
∀u ∈ R .
(2.6)
(2.7)
The operator b : R × L2 ((−r, 0); L2 (Ω)) → L2 (Ω) is a time-dependent external
force with delay; it satisfies:
(I) For all φ ∈ L2 ((−r, 0); L2 (Ω)), the function R 3 t 7→ b(t, φ) ∈ L2 (Ω) is
measurable,
(II) b(t, 0) = 0 for all t ∈ R;
(III) there exists Lb > 0 such that for all t ∈ R and φ1 , φ2 ∈ L2 ((−r, 0); L2 (Ω)),
kb(t, φ1 ) − b(t, φ2 )k ≤ Lb kφ1 − φ2 kL2 ((−r,0);L2 (Ω)) ;
(2.8)
(IV) there exists Cb > 0 such that for all t ≥ τ , and all u, v ∈ L2 ([τ −r, t]; L2 (Ω)),
Z t
Z t
kb(s, us ) − b(s, vs )k2 ds ≤ Cb
ku(s) − v(s)k2 ds .
(2.9)
τ
τ −r
Remark 2.1. From (I)–(III), for T > τ the function R 3 t 7→ b(t, φ) ∈ L2 (Ω) is
measurable and belongs to L∞ ((τ, T ); L2 (Ω)).
4
H. HARRAGA, M. YEBDRI
EJDE-2016/07
The function g ∈ L2loc (R; L2 (Ω)) is an another nondelayed time-dependent external force, u0 ∈ H01 (Ω) is the initial condition in τ and ϕ ∈ L2 ((−r, 0); L2 (Ω)) is
also the initial condition in (τ − r, τ ) , r > 0 is the length of the delay effect.
In this section, we recall some basic concepts about the pullback attractors and
some abstract results about the existence of pullback attractors. Let (Y, d) be a
complete metric space. Let us denote P(Y ) the family of all nonempty subsets of
b = {D(t) : t ∈ R} ⊂
Y , and suppose D is a nonempty class of parameterized sets D
P(Y ).
Definition 2.2 ([11]). A two parameter family of mappings U (t, τ ) : Y → Y ,
t ≥ τ , τ ∈ R, is called to be a norm-to-weak continuous process if
(1) U (τ, τ )x = x for all τ ∈ R, x ∈ Y ;
(2) U (t, s)U (s, τ )x = U (t, τ )x for all t ≥ s ≥ τ , τ ∈ R, x ∈ Y ;
(3) U (t, τ )xn * U (t, τ )x if xn → x in Y .
The following result is useful for satisfying that a process is norm-to-weak continuous.
Proposition 2.3 ([11]). Let Y, Z be two Banach spaces and Y ∗ , Z ∗ be their dual
spaces. Assume that Y is dense in Z, the injection i : Y → Z is continuous and its
adjoint i∗ : Z ∗ → Y ∗ is dense, and {U (t, τ )} is a continuous or weak continuous
process on Z. Then {U (t, τ )} is norm-to-weak continuous on Y if and only if for
t ≥ τ , τ ∈ R, U (t, τ ) maps a compact set of Y to a bounded set of Y .
b = {B(t) : t ∈ R} ∈ D is called
Definition 2.4 ([12]). A family of bounded sets B
b ∈ D,
pullback D-absorbing for the process {U (t, τ )} if for any t ∈ R and for any D
b
there exists τ0 (t, D) ≤ t such that
b .
U (t, τ )D(τ ) ⊂ B(t) for all τ ≤ τ0 (t, D)
Definition 2.5 ([12]). A process {U (t, τ )} is called pullback w-D-limit compact if
b ∈ D, there exists τ0 (t, D)
b ≤ t such that
for all ε > 0 and D
K ∪τ ≤τ0 U (t, τ )D(τ ) ≤ ε ,
where K is the Kuratowski measure of noncompactness of B ∈ P(Y ). This measure
is defined as
K(B) = inf{δ > 0 : B has a finite open cover of sets of diameter less than δ,
and has the following properties.
Lemma 2.6 ([12]). Let B, B0 , B1 be bounded subsets of Y . Then
(1) K(B) = 0 ⇐⇒ K(N (B, ε)) ≤ 2 ⇐⇒ B is compact;
(2) K(B0 + B1 ) ≤ K(B0 ) + K(B1 );
(3) K(B0 ) ≤ K(B1 ) whenever B0 ⊂ B1 ;
(4) K(B0 ∪ B1 ) ≤ max{K(B0 ) , K(B1 )};
(5) K(B) = K(B);
(6) if B is a ball of radius ε then K(B) ≤ 2ε.
b = {A(t) : t ∈ R} ⊂ P(Y ) is said to be a pullback
Definition 2.7 ([12]). A family A
D-attractor for {U (t, τ )} if
(1) A(t) is compact for all t ∈ R;
b is invariant; i.e., U (t, τ )A(τ ) = A(t), for all t ≥ τ ;
(2) A
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
5
b is pullback D-attracting ; i.e.,
(3) A
lim dist(U (t, τ )D(τ ), A(t)) = 0 ,
τ →−∞
b ∈ D and all t ∈ R;
for all D
(4) If {C(t) : t ∈ R} is another family of closed attracting sets then A(t) ⊂ C(t),
for all t ∈ R.
Theorem 2.8 ([12]). Let {U (t, τ )} be a norm-to-weak continuous process such
that {U (t, τ )} is pullback w-D-limit compact. If there exists a family of pullback
b = {B(t) : t ∈ R} ∈ D for the process {U (t, τ )}, then there exists
D-absorbing sets B
a pullback D-attractor {A(t) : t ∈ R} such that
b t) = ∩s≤t ∪τ ≤s U (t, τ )B(τ ) .
A(t) = w(B,
3. Existence of pullback D-attractors
3.1. Existence and uniqueness of weak solutions. First, we define the concept
of weak solution.
Definition 3.1. A function u ∈ L2 ((τ − r, T ); L2 (Ω)) is called a weak solution of
(1.4) if for all T > τ we have
∂u
∈ L2 ((τ, T ); H01 (Ω)) ,
∂t
with u(t) = ϕ(t−τ ) for t ∈ [τ −r, τ ], and for all test functions v ∈ C 1 ([τ, T ]; H01 (Ω))
such that v(T ) = 0, it satisfies
Z T
Z TZ
Z TZ
Z TZ
∂u
−hu, v 0 i +
∇u∇v +
∇ ∇v +
f (u)v
∂t
τ
τ
Ω
τ
Ω
τ
Ω
(3.1)
Z T
Z TZ
u ∈ C([τ, T ]; H01 (Ω))
and
gv + hu0 , v(τ )i .
hb(t, ut ), vi +
=
τ
τ
Ω
Next we have the existence and uniqueness of solutions which are obtained by
the usual Faedo-Galerkin approximation and a compactness method.
Theorem 3.2. For any τ ∈ R, T > τ , u0 ∈ H01 (Ω), ϕ ∈ L2 ((−r, 0); L2 (Ω)) and
0
b
+C
if there exist positive constants η, η 0 < 1/2 such that λ1 > c1 + η+η
2
2η , then
problem (1.4) has a unique weak solution u on (τ, T ).
Proof. Let {ek }k≥1 , be the complete basis of H01 (Ω)∩H 2 (Ω) given by the orthonormal eigenfunctions of −∆ in L2 (Ω). We consider
um (t) =
m
X
γk,m (t)ek ,
m∈N
k=1
which is the approximate solution of Faedo-Galerkin of order m; that is,
dum
∂um
−∆
− ∆um + Pm f (um ) = Pm b(t, um
t ) + Pm g
dt
∂t
um (τ ) = Pm u0 = u0 i.e. Pm um (τ ) → u0 in H01 (Ω)
(3.2)
um (τ + θ) = Pm ϕ(θ) = ϕ(θ) ∀θ ∈ (−r, 0)
for all k ∈ N, where γk,m (t) = hum (t), ek i denote the Fourier coefficients such
0
that γm,k ∈ C 1 ((τ, T ); R) ∩ L2 ((τ − r, T ), R) , γk,m
(t) is absolutely continuous, and
6
H. HARRAGA, M. YEBDRI
EJDE-2016/07
Pm
Pm u(t) = k=1 hu, ek iek is the orthogonal projection of u ∈ L2 (Ω) or u ∈ H01 (Ω)
in Hm = span{e1 , . . . , em }.
It is well-known that the above finite-dimensional delayed system is well-posed
at least locally (see for example [6, Theorem 2.1, p. 14]). Indeed; for fixed m, the
system (3.2) defines a linear system of differential equations on Rm . Then we can
apply differential equations theory for local existence and uniqueness of solutions
to the system (3.2), i.e. for initial conditions (ϕ, v m (τ )) in L2 ((−r, 0); Rm ) × Rm ,
there exist tm > 0 and a unique solution of (3.2) v m (t) = (γ1,m (t) . . . γm,m (t))T with
v m ∈ L2 ((τ − r, τ ); Rm ) such that v m |[τ −r,τ ] = ϕ and v m (τ ) = a, and v m |[τ,tm ] ∈
C 1 ([τ, tm ]; Rm ). Hence, the solution of (3.2) is defined on the interval [τ, tm ] with
τ < tm < T . The a priori estimates for the Faedo-Galerkin approximate solutions
that we obtain will show that tm = T .
Claim 3.3. {um } is bounded in L∞ ((τ, T ); H01 (Ω)).
Multiplying (3.2) by um and integrating over Ω, we obtain
Z
Z
1 d
m
m
(kum (t)k2 + k∇um (t)k2 ) + k∇um (t)k2 + f (um )um = (b(t, um
t )u + gu ) .
2 dt
Ω
Ω
Using (2.1) and the Cauchy inequality, we obtain
1 d
(kum (t)k2 + k∇um (t)k2 ) + k∇um (t)k2 − c1 kum (t)k2 − c2 |Ω|
2 dt
1
η m
1
η0 m
2
2
2
≤
kb(t, um
kg(t)k
+
ku (t)k2 .
t )k + ku (t)k +
2η
2
2η 0
2
As
λ1 kuk2 ≤ k∇uk2 ,
(3.3)
then
d
(kum (t)k2 + k∇um (t)k2 ) + (2λ1 − 2c1 − η − η 0 )kum (t)k2
dt
1
1
2
2
≤ kb(t, um
t )k + 0 kg(t)k + c2 |Ω| .
η
η
Integrating this estimate over [τ, t], t ≤ T , we find that
Z t
kum (t)k2 + k∇um (t)k2 + (2λ1 − 2c1 − η − η 0 )
kum (s)k2
τ
Z
Z
1 t
1 t
m
2
m
2
m 2
≤ ku (τ )k ds + k∇u (τ )k +
kb(s, us )k ds + 0
kg(s)k2 ds
η τ
η τ
+ c2 |Ω|(t − τ ) .
Therefore, by using (2.9), we obtain
m
2
m
2
0
Z
t
ku (t)k + k∇u (t)k + (2λ1 − 2c1 − η − η )
kum (s)k2 ds
τ
Z
Cb τ
≤ kum (τ )k2 + k∇um (τ )k2 +
kum (s)k2 ds
η τ −r
Z
Z
Cb t m
1 t
2
+
ku (s)k ds + 0
kg(s)k2 ds + c2 |Ω|(t − τ ) .
η τ
η τ
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DIFFUSION EQUATIONS WITH DELAY
7
So, one has
Cb kum (t)k2 + k∇um (t)k2 + 2λ1 − 2c1 − η − η 0 −
η
Z τ
C
b
≤ kum (τ )k2 + k∇um (τ )k2 +
kum (s)k2 ds
η τ −r
Z
1 t
+ 0
kg(s)k2 ds + c2 |Ω|(t − τ ) .
η τ
Hence, when 2λ1 − 2c1 − η − η 0 −
Cb
η
t
kum (s)k2
τ
(3.4)
> 0 and g ∈ L2loc (R; L2 (Ω)), one gets
k∇um (t)k2 ≤ kum (τ )k2 + k∇um (τ )k2 +
+
Z
Cb
kϕk2L2 ((−r,0);L2 (Ω))
η
1
kgk2L2 ([τ,t];L2 (Ω)) + c2 |Ω|(t − τ ) .
η0
(3.5)
By this estimate, for all T > τ , we arrive at
{um } is bounded in L∞ ((τ, T ); H01 (Ω))
(3.6)
Then we deduce that the local solution um can be extended to the interval [τ, T ].
∂ m
Claim 3.4. ∂t
u
is bounded in L2 ((τ, T ); H01 (Ω)).
∂um
∂t
and integrating over Ω, we have
Z
d
1 d
∂um
d
k um (t)k2 + k∇ um (t)k2 +
k∇um (t)k2 +
f (um )
dt
dt
2 dt
∂t
Ω
Z
Z
m
m
∂u
∂u
=
b(t, um
+
g
.
t )
∂t
∂t
Ω
Ω
Multiplying (3.2) by
(3.7)
As
d
dt
Z
Z
F (u) =
Ω
f (u)
Ω
∂u
,
∂t
(3.7) becomes
d
1 d
d
k∇um (t)k2 + 2
k um (t)k2 + k∇ um (t)k2 +
dt
dt
2 dt
Z
Z
∂um
∂um
=
b(t, um
)
+
g
.
t
∂t
∂t
Ω
Ω
Z
F (um )
Ω
Using the Young inequality, we find
d
1 d
k∇ um (t)k2 +
(k∇um (t)k2 + 2
dt
2 dt
Z
Ω
F (um )) ≤
1
1
2
2
kb(t, um
t )k + kg(t)k .
2
2
Integrating over [τ, t], t ≤ T we obtain
Z t
Z
d
1
k∇ um (s)k2 ds +
k∇um (t)k2 + 2
F (um (t, x))
ds
2
τ
Ω
Z
1Z t
1
2
k∇um (τ )k2 + 2
F (um (τ, x)) +
kb(s, um
≤
s )k ds
2
2 τ
Ω
Z
1 t
+
kg(s)k2 ds .
2 τ
8
H. HARRAGA, M. YEBDRI
EJDE-2016/07
By (2.9), we have
Z t
Z
d
1
k∇ um (s)k2 ds +
k∇um (t)k2 + 2
F (um (t, x))
ds
2
τ
Ω
Z t
Z
Z
1 t
1
C
b
kum (s)k2 ds +
kg(s)k2 ds
≤ (k∇um (τ )k2 + 2
F (um (τ, x))) +
2
2 τ −r
2 τ
Ω
Z
C Z t
1
b
k∇um (τ )k2 + 2
F (um (τ, x)) +
kum (s)k2 ds
≤
2
2
Ω
τ
Z
Cb τ
1
m
2
2
+
ku (s)k + kgkL2 ([τ,t];L2 (Ω)) .
2 τ −r
2
From this estimate and (3.4), we deduce that, for all T > τ ,
∂ m
u
is bounded in L2 ((τ, T ); H01 (Ω)) .
∂t
(3.8)
Lemma 3.5 ([16, Lemma 3.1]). If {um } is bounded in L∞ ((τ, T ), H01 (Ω)), then
{f (um )} is bounded in Lq ((τ, T ); Lq (Ω)) ,
(3.9)
where q = (2N + 4)/(αN ).
By (3.6), (3.8), (3.9), hypothesis (IV) and remark (2.1), we can extract a subsequence (relabeled the same) such that
um * u weakly* in L∞ ((τ, T ); H01 (Ω)),
∆u
m
2
* ∆u weakly in L ((τ, T ); H
−1
(Ω)),
∂um
∂u
*
weakly in L2 ((τ, T ); H01 (Ω)),
∂t
∂t
∂um
∂u
∆(
) * ∆( ) weakly in L2 ((τ, T ); H −1 (Ω)),
∂t
∂t
f (um ) * σ 0 weakly in Lq ((τ, T ); Lq (Ω)),
b(., um
. )
→ b(., u. )
2
2
strongly in L ((τ, T ); L (Ω)) .
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
By (3.10), we have that um * u weakly in L2 ((τ, T ); L2 (Ω)) (L∞ ((τ, T ); H01 (Ω)) ⊂
m
∂u
2
2
L2 ((τ, T ); L2 (Ω))), and by (3.12), we have ∂u
∂t * ∂t weakly in L ((τ, T ); L (Ω))
2
2
2
1
(L ((τ, T ); H0 (Ω)) ⊂ L ((τ, T ); L (Ω))). So, we can extract a subsequence u of um
that satisfies
um → u strongly in L2 ((τ, T ); L2 (Ω)) . Thus um → u a.e [τ, T ] × Ω .
(3.16)
By (3.16) and the fact that f is continuous, we deduce that f (um ) → f (u) a.e
[τ, T ] × Ω, So, from (3.14) and [13, Lemma 1.3, p. 12] we can identify σ 0 with f (u).
Now, we have to prove that u(τ ) = u0 . Recall that (3.1),
Z T
Z TZ
Z TZ
Z TZ
∂u
−hu, v 0 i +
∇ ∇v +
∇u∇v +
f (u)v
∂t
τ
τ
Ω
τ
Ω
τ
Ω
(3.17)
Z T
Z TZ
=
hb(t, ut ), vi +
gv + hu(τ ), v(τ )i .
τ
τ
Ω
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
In a similar way, from the Faedo-Galerkin approximations, we have
Z TZ
Z T
Z TZ
Z TZ
∂um
∇
−hum , v 0 i +
f (um )v
∇um ∇v +
∇v +
∂t
τ
Ω
τ
τ
Ω
τ
Ω
Z TZ
Z T
gv + hum (τ ), v(τ )i .
hb(t, um
),
vi
+
=
t
τ
τ
(3.18)
Ω
Using the fact that um (τ ) → u0 in H01 (Ω) and (3.10)-(3.15) we find that
Z T
Z TZ
Z TZ
Z TZ
∂u
0
−hu, v i +
∇ ∇v +
∇u∇v +
f (u)v
∂t
τ
τ
Ω
τ
Ω
τ
Ω
Z T
Z TZ
=
hb(t, ut ), vi +
gv + hu0 , v(τ )i .
τ
9
τ
(3.19)
Ω
Since v(τ ) is arbitrarily, comparing (3.17) and (3.19) we deduce that u(τ ) = u0 .
To prove that u ∈ C([τ, T ]; H01 (Ω)), we put wm = um − u then we have
∂
∂ m
w − ∆ wm − ∆wm + f (um ) − f (u) = b(t, um
t ) − b(t, ut ) .
∂t
∂t
Multiplying this equation by wm and integrating over Ω, we obtain
Z
d
m
2
m
2
m
2
(kw (t)k + k∇w (t)k ) + 2k∇w (t)k + 2 (f (um ) − f (u))wm
dt
Ω
Z
m
m
= 2 (b(t, ut ) − b(t, ut ))(u − u) .
Ω
By (2.2), (2.8) and Young’s inequality, we obtain
d
(kwm (t)k2 + k∇wm (t)k2 ) + 2k∇wm (t)k2
dt
≤ (2c3 + Lb )kwm (t)k2 + Lb kwtm k2L2 ((−r,0);L2 (Ω)) .
Hence, by (3.3), one gets
d
(kwm (t)k2 + k∇wm (t)k2 ) ≤ 2c3 kwm (t)k2 + Lb
dt
≤
Z
0
kwm (t + θ)k2 dθ
−r
2c3 + Lb
k∇wm (t)k2 + 2Lb
λ1
Z
0
kwm (t + θ)k2 dθ .
−r
Integrating over [τ, t], we obtain
kwm (t)k2 + k∇wm (t)k2 − kwm (τ )k2 + k∇wm (τ )k2
Z
Z tZ 0
2c3 + Lb t
m
2
≤
k∇w (s)k + Lb
kwm (s + θ)k2 dθds
λ1
τ
τ
−r
Z
Z 0Z t
2c3 + Lb t
k∇wm (s)k2 + Lb
kwm (s)k2 dsdθ
≤
λ1
τ
−r τ −r
Z
Z τ
Z t
2c3 + Lb t
≤
k∇wm (s)k2 + Lb r
kwm (s)k2 ds + Lb r
kwm (s)k2 ds .
λ1
τ
τ −r
τ
2c3 +Lb
Taking β > max
, Lb r , we obtain
λ1
kwm (t)k2 + k∇wm (t)k2 ≤ kwm (τ )k2 + k∇wm (τ )k2
10
H. HARRAGA, M. YEBDRI
Z
τ
kwm (s)k2 ds + β
+ Lb r
τ −r
EJDE-2016/07
t
Z
(k∇wm (s)k2 + kwm (s)k2 )ds .
τ
Applying the Gronwall lemma to this estimate, we obtain
kwm (t)k2 + k∇wm (t)k2
Z
≤ kwm (τ )k2 + k∇wm (τ )k2 + Lb r
0
kwm (τ + θ)k2 dθ eβ(t−τ ) .
(3.20)
−r
m
0
m
Since u (τ ) → u and u (τ + θ) → ϕ(θ), the estimate (3.20) shows that um → u
uniformly in C([τ, T ]; H01 (Ω)).
By concatenation of solutions, it is clear that we obtain at least one global weak
solution to (1.4) defined on (τ, +∞).
Finally, we prove the uniqueness and continuous dependence of the solution on
the data. To do this, we consider u1 , u2 two solutions of (1.4) with the same initial
conditions u0 and ϕ. Let w = u1 − u2 , and similarly as in the proof of (3.20), we
have
kw(t)k2 + k∇w(t)k2
Z 0
(3.21)
2
2
≤ kw(τ )k + k∇w(τ )k + 2Lb r
kw(τ + θ)k2 dθ eβ(t−τ ) ,
−r
and this completes the proof of the theorem because w(τ ) = 0, and w(τ +θ) = 0. 3.2. Pullback D-attractors. Invoking Theorem 3.2, we will apply the above results in the phase space X := H01 (Ω) × L2 ((−r, 0); L2 (Ω)), which is a Hilbert space
with the norm
Z
0
k(u0 , ϕ)k2X = k∇u0 k2 +
kϕ(θ)k2 dθ ,
−r
with a pair (u0 , ϕ) of X. First, we give the following consequence the theorem of
the existence and uniqueness.
Proposition 3.6. We consider g ∈ L2loc (R; L2 (Ω)), b : R × L2 ((−r, 0); L2 (Ω)) →
L2 (Ω) with the hypotheses (I)–(IV) and f ∈ C 1 (R; R) satisfying (2.1)–(2.5). Then
the family of mappings U (t, τ ) : X → X,
(u0 , ϕ) 7−→ U (t, τ )(u0 , ϕ) = (u(t), ut ) ,
(3.22)
2
with (t, τ ) ∈ R and u the weak solution to (1.4), defines a continuous process.
Next, we need to consider the Hilbert space
X1 = H01 (Ω) × L2 ((−r, 0); H01 (Ω)) ,
with the norm
0
k(u
, ϕ)k2X1
0 2
Z
0
= k∇u k +
k∇ϕ(θ)k2 dθ .
−r
We remark that when t − τ ≥ r, U (t, τ ) maps X to X1 . To prove a pullback
-absorbing set for the process U (t, τ ), we need the following lemma.
Lemma 3.7. Assume that f satisfies (2.3), (2.6) and (2.7). For all u, v ∈ L2 ((τ −
r, t); L2 (Ω)), b and g satisfy
Z t
Z t
eσs kb(s, us ) − b(s, vs )k2 ds ≤ Cb
eσs ku(s) − v(s)k2 ds ,
(3.23)
τ
τ −r
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
11
and
Z
t
eσs kg(s)k2 ds < ∞ ,
∀t ∈ R ,
(3.24)
−∞
1
where 0 < σ < δ 0 < min 2c4 , 2λ2λ
. Then for all t for which t ≥ τ + r and all
1 +1
(u0 , ϕ) ∈ X, we have the estimates
Z t
o
2N
N −2
eσs kg(s)k2 ds ,
(3.25)
k∇u(t)k2 ≤ c e−σ(t−τ ) k(u0 , ϕ)kX
+ 1 + e−σt
−∞
Z
t
k∇u(s)k2 ds
t−r
−1
≤ cµ
Z
n
2N
N −2
−σ(t−τ −r)
0
σr −σ(t−r)
e
k(u , ϕ)kX + e e
t
(3.26)
o
e kg(s)k ds ,
σs
2
−∞
0
where µ := 4(δ − σ −
Cb
λ1 )
> 0.
Proof. Multiplying (1.4) by u +
∂u
∂t
and integrating over Ω, we obtain
Z
1 d
d
∂u
2
2
2
2
(ku(t)k + 2k∇u(t)k ) + k u(t)k + k∇u(t)k +
f (u)(u +
)
2 dt
dt
∂t
Ω
d
+ k∇ u(t)k2
dt
Z
Z
∂u
∂u
=
b(t, ut )(u +
)+
g(u +
).
∂t
∂t
Ω
Ω
By (2.6), the Cauchy-Schwarz and Young inequalities, one gets
Z
Z
1 d
(ku(t)k2 + 2k∇u(t)k2 + 2
F (u(t))) + k∇u(t)k2 + c4
F (u)
2 dt
Ω
Ω
≤ kb(t, ut )k2 + kg(t)k2 + (1 + δ)ku(t)k2 + cδ |Ω| .
By (3.3), we obtain
Z
Z
2(1 + δ)
d
(ku(t)k2 + 2k∇u(t)k2 + 2
F (u(t))) + (2 −
)k∇u(t)k2 + 2c4
F (u)
dt
λ1
Ω
Ω
≤ 2kb(t, ut )k2 + 2kg(t)k2 + 2cδ |Ω| .
We can choose δ small enough such that
2−
2(1 + δ) k∇u(t)k2 ≥ δ 0 (ku(t)k2 + 2k∇u(t)k2 ) ,
λ1
(3.27)
1
where δ 0 < min{2c4 , 2λ2λ
}. So, we can write
1 +1
d
γ1 (t) + δ 0 γ1 (t) ≤ 2kg(t)k2 + 2kb(t, ut )k2 + 2cδ |Ω| ,
dt
(3.28)
where
γ1 (t) = ku(t)k2 + 2k∇u(t)k2 + 2
Z
F (u(t)) .
(3.29)
Ω
Multiplying (3.28) by eσt such that 0 < σ < δ 0 , we obtain
eσt
d
γ1 (t) + δ 0 eσt γ1 (t) ≤ 2eσt (kg(t)k2 + kb(t, ut )k2 + cδ |Ω|) ,
dt
(3.30)
12
H. HARRAGA, M. YEBDRI
EJDE-2016/07
whereupon
d σt
(e γ1 (t)) ≤ (σ − δ 0 )eσt γ1 (t) + 2eσt (kg(t)k2 + kb(t, ut )k2 + cδ |Ω|) .
dt
Integrating this last estimate over [τ, t], we find that
γ1 (t)
−σ(t−τ )
0
−σt
Z
t
eσs γ1 (s)ds
γ1 (τ ) + (σ − δ )e
τ
Z t
Z t
Z t
σs
2
−σt
σs
2
−σt
−σt
eσs ds
e kb(s, us )k ds + 2e cδ |Ω|
e kg(s)k ds + 2e
+ 2e
τ
τ
τ
Z t
Z t
σs
−σt
−σ(t−τ )
0 −σt
eσs kg(s)k2 ds
e γ1 (s)ds + 2e
≤e
γ1 (τ ) + (σ − δ )e
τ
τ
Z t
+ 2e−σt
eσs kb(s, us )k2 ds + 2e−σt cδ |Ω|σ −1 (1 − e−σ(t−τ ) )
τ
Z t
≤ e−σ(t−τ ) γ1 (τ ) + (σ − δ 0 )e−σt
eσs γ1 (s)ds
τ
Z t
Z t
+ 2e−σt
eσs kg(s)k2 ds + 2e−σt
eσs kb(s, us )k2 ds + 2e−σt cδ |Ω|σ −1 .
≤e
τ
τ
Therefore, using (3.23) and (II), one has
−σ(t−τ )
0
−σt
Z
t
Z
t
γ1 (τ ) + (σ − δ )e
eσs γ1 (s)ds
τ
Z t
Z t
−σt
σs
2
−σt
+ 2e
e kg(s)k ds + 2Cb e
eσs ku(s)k2 ds + 2cδ |Ω|σ −1
γ1 (t) ≤ e
τ −r
τ
≤ e−σ(t−τ ) γ1 (τ ) + (σ − δ 0 )e−σt
eσs γ1 (s)ds
τ
Z τ
Z t
+ 2Cb e−σ(t−τ )
ku(s)k2 ds + 2Cb e−σt
eσs ku(s)k2 ds
τ −r
+ 2e−σt
τ
t
Z
eσs kg(s)k2 ds + 2cδ |Ω|σ −1 .
τ
(3.31)
We use 2.7 in (3.29) to obtain
γ1 (t) ≥ ku(t)k2 + 2k∇u(t)k2 − 2δku(t)k2 − 2c0δ |Ω|
≥ (1 − 2δ)ku(t)k2 + 2k∇u(t)k2 − 2c0δ |Ω|
1
≥ k∇u(t)k2 .
2
(3.32)
On the other hand,
2
Z
2
γ1 (τ ) = ku(τ )k + 2k∇u(τ )k + 2
F (u(τ )) .
Ω
By (2.3), we have
Z
Z
k
F (u) ≤ k
|u| +
α
+
1
Ω
Ω
Z
Ω
|u|α+1 .
(3.33)
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
13
Using the Holder inequality and the fact that α + 1 ≤ N2N
−2 , one has
Z
Z
Z
p
2N
c
F (u) ≤ k |Ω|( |u|2 )1/2 +
|u| N −2
α+1 Ω
Ω
Ω
2N
p
c
≤ k |Ω|kuk +
.
kuk N −2
2N
α+1
L N −2 (Ω)
2N
By the embedding of H01 (Ω) in L N −2 (Ω), (3.3) and the fact that 1 <
Z
p
2N
c
|Ω|k∇uk +
k∇uk N −2
F (u) ≤ kλ−1
1
α+1
Ω
p
2N
2N
c
−1
N
−2
+
|Ω|k∇uk
k∇uk N −2
≤ cλ1
α+1
p
2N
c
−1
≤ (cλ1
|Ω| +
)k∇uk N −2
α+1
2N
N −2 ,
2N
≤ ck∇uk N −2 ,
we have
(3.34)
where c is a positive constant. Using this inequality in (3.33), we obtain
2N
γ1 (τ ) ≤ ku(τ )k2 + 2k∇u(τ )k2 + ck∇u(τ )k N −2 .
Using (3.3) and 2 <
2N
N −2 ,
one finds that
2N
2N
2
N −2 ≤ ck∇u(τ )k N −2 .
γ1 (τ ) ≤ (λ−1
1 + 2)k∇u(τ )k + ck∇u(τ )k
(3.35)
We substitute (3.32) and (3.35) in (3.31), one gets
Z τ
2N
1
2
−σ(t−τ )
−σ(t−τ )
N
−2
k∇u(t)k ≤ ce
k∇u(τ )k
+ 2Cb e
ku(s)k2 ds
2
τ −r
Z t
Z
+ (σ − δ 0 )e−σt
eσs ku(s)k2 + 2k∇u(s)k2 + 2
F (u(s)) ds
τ
Ω
Z t
Z t
+ 2e−σt
eσs kg(s)k2 ds + 2Cb e−σt
eσs ku(s)k2 ds + 2cδ |Ω|σ −1 .
τ
τ
So, by (3.3),
2N
k∇u(t)k2 ≤ ce−σ(t−τ ) k∇u(τ )k N −2 + 4Cb e−σ(t−τ )
Z
τ
ku(s)k2 ds
τ −r
0
−σt
Z
t
σs
2
2
Z
+ 2(σ − δ )e
e (ku(s)k + 2k∇u(s)k + 2
F (u(s)))ds
τ
Ω
Z t
Z t
4Cb −σt
e
eσs k∇u(s)k2 ds + 4cδ |Ω|σ −1 .
+ 4e−σt
eσs kg(s)k2 ds +
λ1
τ
τ
Then, for δ 0 − σ −
Cb
λ1
> 0 and the fact that 2 < N2N
−2 , we have
Z t
Cb −σt
k∇u(t)k2 + 4 δ 0 − σ −
e
eσs k∇u(s)k2 ds
λ1
τ
2N
2N
≤ ce−σ(t−τ ) k∇u(τ )k N −2 + ce−σ(t−τ ) kϕkLN2−2
((−r,0);L2 (Ω))
Z t
+ 4e−σt
eσs kg(s)k2 ds + 4cδ |Ω|σ −1
τ
n
2N
2N
≤ c e−σ(t−τ ) (k∇u0 k N −2 + kϕkLN2−2
((−r,0);L2 (Ω)) )
14
H. HARRAGA, M. YEBDRI
+ 1 + e−σt
t
Z
EJDE-2016/07
o
eσs kg(s)k2 ds .
τ
By using (3.24), we have
Z
Cb −σt t σs
k∇u(t)k + 4 δ − σ −
e k∇u(s)k2 ds
e
λ1
τ
n
2N
2N
≤ c e−σ(t−τ ) (k∇u0 k N −2 + kϕkLN2−2
((−r,0);L2 (Ω)) )
Z t
o
eσs kg(s)k2 ds .
+ 1 + e−σt
0
2
(3.36)
−∞
Whereupon, for all t ≥ τ , we obtain (3.25), and
Z t
eσs k∇u(s)k2 ds
µe−σt
τ
Z
n
2N
N −2
−σ(t−τ )
≤c e
k(u0 , ϕ)kX
+ 1 + e−σt
(3.37)
t
o
eσs kg(s)k2 ds ,
−∞
Cb
λ1
0
> 0. Furthermore, for τ ≤ t − r, we have
where µ := 4 δ − σ −
Z t
Z t
Z t
eσs k∇u(s)k2 ds ≥
eσs k∇u(s)k2 ds ≥ eσ(t−r)
k∇u(s)k2 ds ,
τ
t−r
t−r
as [t − r, t] ⊂ [τ, t]. Hence, (3.37) becomes
Z t
−σr
µe
k∇u(s)k2 ds
t−r
Z
n
2N
N −2
≤ c e−σ(t−τ ) k(u0 , ϕ)kX
+ 1 + e−σt
t
o
eσs kg(s)k2 ds .
−∞
Therefore, for all t ≥ τ + r, we obtain (3.26), and this completes the proof.
Let R be the set of all functions ρ : R → (0, +∞) such that
2N
lim eσt ρ N −2 (t) = 0 .
t→−∞
b = {D(t) : t ∈ R} ⊂ P(X) such that
By D we denote the class of all families D
D(t) ⊂ BX (0, ρ(t)), for some ρ ∈ R, where BX (0, ρ(t)) denotes the closed ball in
X centered at 0 with radius ρ(t). Let
Z t
−σt
ρ1 (t) = c 1 + e
eσs kg(s)k2 ds ,
−∞
0
and R(t) ≥ 0, R (t) ≥ 0, where
R2 (t) = (1 + µ−1 eσr )ρ1 (t) ,
N
R02 (t) = cρ1N −2 (t) + cρ1 (t) + ckgk2L2 ([t−r,t];L2 (Ω)) .
Lemma 3.8 (Pullback D-absorbing set). Under the assumptions of Lemma 3.7,
b given by
the family B
n
o
dφ
B(t) = (v 0 , φ) ∈ X1 : k(v 0 , φ)kX1 ≤ R(t), k kL2 ((−r,0);L2 (Ω)) ≤ R0 (t) , (3.38)
ds
is pullback D-absorbing for the process U (., .) defined by (3.22) .
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
15
Proof. First, we observe that for all t ∈ R,
B(t) ⊂ {(v 0 , φ) ∈ X : k(v 0 , φ)kX ≤ R(t)} ,
(3.39)
with
2N
lim eσt R N −2 (t) = 0 ,
t→−∞
b ∈ D.
and so B
Now, we prove that U (t, τ )D(τ ) ⊂ B(t), for all τ ≤ τ0 . To do this, we proceed
in two steps.
Step 1. This step concerns the asymptotic estimate using R(t) for t ∈ R, fixed. It
may be proved as follows. By definition, we have
Z t
0
2
2
k∇u(s)k2 ds .
(3.40)
kU (t, τ )(u , ϕ)kX1 = k∇u(t)k +
t−r
From (3.26), for any t − r ≥ τ , we have
Z t
2N
N −2
k∇u(s)k2 ds ≤ cµ−1 e−σ(t−τ −r) k(u0 , ϕ)kX
t−r
+ µ−1 eσr c(1 + e−σt
Z
(3.41)
t
eσs kg(s)k2 ds) ,
−∞
0
for any (u , ϕ) ∈ X. We substitute this inequality and (3.25) in (3.40); by the
definition of ρ1 (t), we obtain
2N
N −2
(1 + µ−1 eσr ) + (1 + µ−1 eσr )ρ1 (t)
kU (t, τ )(u0 , ϕ)k2X1 ≤ ce−σ(t−τ ) k(u0 , ϕ)kX
2N
N −2
≤ ce−σ(t−τ ) k(u0 , ϕ)kX
(1 + µ−1 eσr ) + R2 (t) ,
for all t − r ≥ τ and all (u0 , ϕ) ∈ X. Hence
kU (t, τ )(u0 , ϕ)k2X1 ≤ R2 (t) ,
as e
στ
(3.42)
→ 0 when τ → −∞.
Step 2. This step concerns the asymptotic estimate using R0 (t). We assume that
t − 2r ≥ τ . Multiplying (1.4) by ∂u
∂t and integrating over Ω, we obtain
Z
d
1 d
1
1
k u(t)k2 +
(k∇u(t)k2 + 2
F (u)) ≤ kb(t, ut )k2 + kg(t)k2 .
dt
2 dt
2
2
Ω
Integrating over [t − r, t], we find
Z t
Z
d
1
k u(s)k2 ds +
k∇u(t)k2 + 2
F (u(t))
2
t−r ds
Ω
Z
1
≤ (k∇u(t − r)k2 + 2
F (u(t − r)))
2
Ω
Z
Z
1 t
1 t
+
kg(s)k2 ds +
kb(s, us )k2 ds .
2 t−r
2 t−r
From (II), (IV), and (3.3), one has
Z t
Z t
Z
kb(s, us )k2 ds ≤ Cb
ku(s)k2 ds ≤ Cb λ−1
1
t−r
t−2r
t
t−2r
k∇u(s)k2 ds .
(3.43)
16
H. HARRAGA, M. YEBDRI
EJDE-2016/07
By this estimate and (3.43),
Z t
Z
d
1
k u(s)k2 ds ≤ (k∇u(t − r)k2 + 2
F (u(t − r)))
2
t−r ds
Ω
Z
Z t
Cb λ−1
1 t
2
1
kg(s)k ds +
k∇u(s)k2 ds .
+
2 t−r
2
t−2r
By (3.34) and the fact that 2 < N2N
−2 , we have
Z t
2N
d
1
k u(s)k2 ds ≤ (k∇u(t − r)k2 + 2ck∇u(t − r)k N −2 )
ds
2
t−r
Z
Z t
Cb λ−1
1 t
2
1
kg(s)k ds +
k∇u(s)k2 ds
+
2 t−r
2
t−2r
2N
≤ ck∇u(t − r)k N −2 + ckgk2L2 ([t−r,t];L2 (Ω))
Z t
+c
k∇u(s)k2 ds .
(3.44)
t−2r
Now, we estimate k∇u(t − r)k2 . Replacing t by t − r in (3.25), we obtain
2N
N −2
k∇u(t − r)k2 ≤ ce−σ(t−r−τ ) k(u0 , ϕ)kX
Z t−r
+ c 1 + e−σ(t−r)
eσ(s−r) kg(s − r)k2 ds .
−∞
Because t − r ≤ t and eσr > 1, we have
2N
N −2
k∇u(t − r)k2 ≤ ceσr e−σ(t−τ ) k(u0 , ϕ)kX
Z t
+ eσr c(1 + e−σt
eσs kg(s)k2 ds)
−∞
σr −σ(t−τ )
≤ ce e
2N
N −2
+ eσr ρ1 (t) .
k(u , ϕ)kX
0
Hence
NN−2
2N
2N
N −2
k∇u(t − r)k N −2 ≤ ceσr e−σ(t−τ ) k(u0 , ϕ)kX
+ eσr ρ1 (t)
.
Since
N
N −2
> 1, using a convexity argument,
2N
N
2N 2
2
N
(N −2)
k∇u(t − r)k N −2 ≤ ce− N −2 σ(t−τ ) k(u0 , ϕ)kX
+ cρ1N −2 (t) .
Using (3.26), and taking 2r in place of r, we have
Z t
2N
N −2
k∇u(s)k2 ds ≤ ce−σ(t−τ ) k(u0 , ϕ)kX
+ cρ1 (t) ,
(3.45)
(3.46)
t−2r
for any t − 2r ≥ τ . From (3.44)-(3.46) we conclude that for all t − 2r ≥ τ and all
(u0 , ϕ) ∈ X,
Z t
2N 2
2N
N
d
(N −2)2
N −2
k u(s)k2 ds ≤ ce− N −2 σ(t−τ ) k(u0 , ϕ)kX
+ ce−σ(t−τ ) k(u0 , ϕ)kX
t−r ds
N
+ cρ1N −2 (t) + cρ1 (t) + ckgk2L2 ([t−r,t];L2 (Ω)) .
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
17
Hence, for all t − 2r ≥ τ and for any (u0 , ϕ) ∈ X, we have
Z t
d
k u(s)k2 ds
ds
t−r
2N
N
2N 2
2
(N −2)
N −2
+ ce− N −2 σ(t−τ ) k(u0 , ϕ)kX
+ R02 (t) ,
≤ ce−σ(t−τ ) k(u0 , ϕ)kX
so we obtain
Z
t
d
u(s)k2 ds ≤ R02 (t) ,
(3.47)
ds
t−r
as eστ → 0 when τ → −∞. Then, it is clear to see from (3.42), (3.47) and the
b given by (3.38) is pullback D-absorbing for the
definition of D that the family B
process U (., .).
k
In what follows, we need the following result.
Proposition 3.9. Let {U (t, τ )} be a process on X, and let {B(t) : t ∈ R} be a
b ∈ D and
pullback D-absorbing set of {U (t, τ )}. Suppose that for each t ∈ R, any B
b ε) ≤ t and δ > 0 such that
any ε > 0, there exist τ0 = τ0 (t, B,
(1) for all τ ≤ τ0 and (u(t), ut ) ∈ U (t, τ )B(τ ), kP (u(t), ut )kX1 is bounded;
(2) for all τ ≤ τ0 and (u(t), ut ) ∈ U (t, τ )B(τ ), k(I − P )(u(t), ut )kX1 < ε;
(3) for all τ ≤ τ0 , ut ∈ U (t, τ )B(τ ) and all l ∈ R with |l| < δ, we have
kP (Tl ut − ut )kL2 ((−r,0);L2 (Ω)) < ε ,
where Tl ut is the translation (Tl ut )(θ) = u(t + θ + l) with θ ∈ (−r, 0) and P
is the canonical projector on the finite dimensional subspace Vn of H01 (Ω)
and L2 (Ω), and I is the identity. Then {U (t, τ )} is pullback ω-D-limit
compact in X with respect to each t ∈ R.
Proof. (i) First, we prove that {U (t, τ )} is pullback ω-D-limit compact in X1 . Note
that by (2) in the Lemma 2.6, one has
K ∪τ ≤τ0 U (t, τ )B(τ ) ≤ K P ∪τ ≤τ0 U (t, τ )B(τ )
(3.48)
+ K (I − P )(∪τ ≤τ0 U (t, τ )B(τ )) .
Assumption (1) gives that {P ∪τ ≤τ0 U (t, τ )B(τ )} is contained in a ball of finite
radius. So by (3) in Lemma 2.6, we obtain
K P ∪τ ≤τ0 U (t, τ )B(τ ) ≤ K(B(0, ε0 )) ,
(3.49)
and by (6) in Lemma 2.6, we obtain
K(B(0, ε0 )) ≤ 2ε0 .
(3.50)
Thus, by (3.49) and (3.50) it follows that
K P ∪τ ≤τ0 U (t, τ )B(τ ) ≤ 2ε0 .
(3.51)
On the other hand, assumption (2) and property (6) in Lemma 2.6 give
K (I − P ) ∪τ ≤τ0 U (t, τ )B(τ ) ≤ 2ε .
(3.52)
Therefore, by (3.48), (3.51) and (3.52) we deduce that
K ∪τ ≤τ0 U (t, τ )B(τ ) ≤ 2ε0 ,
18
H. HARRAGA, M. YEBDRI
EJDE-2016/07
where ε0 := ε0 + ε, and this shows that {U (t, τ )} is pullback D-w-limit compact
0
0
0
0
in X1 , i.e.; for all τ ≤ τ0 , any sequences τ n → −∞ and (u0,n , ϕn ) ∈ B(τ n ), the
0
0
0
0
0
sequence {(un (t), unt )} = {U (t, τ n )(u0,n , ϕn )} is relatively compact in X1 .
(ii) Second, we will check the equicontinuity property of ut in L2 ((−r, 0); L2 (Ω)).
To this end, we need to use the Lp -version of Arzelà-Ascoli theorem (see [3, theorem
IV.25, p.72]).
Assumption (2) gives
k(I − P )ut kL2 ((−r,0);H01 (Ω)) < ε .
(3.53)
Since H01 (Ω) ,→ L2 (Ω) with continuous injection, we have
k(I − P )ut kL2 ((−r,0);L2 (Ω)) ≤ c5 k(I − P )ut kL2 ((−r,0);H01 (Ω)) .
So by this estimate and (3.53), one has
k(I − P )ut kL2 ((−r,0);L2 (Ω)) < ε0 ,
(3.54)
0
where ε := c5 ε. From (3.54) and assumption (3), we deduce that for all τ ≤ τ0 ,
ut ∈ U (t, τ )B(τ ) and all l ∈ R+ with l < δ, we have
kTl ut − ut kL2 ((−r,0);L2 (Ω))
≤ kP (Tl ut − ut )kL2 ((−r,0);L2 (Ω)) + k(I − P )(Tl ut − ut )kL2 ((−r,0);L2 (Ω)) < ε00 ,
with ε00 := ε + ε0 and this is the desired equicontinuity.
0
From (i), we deduce that {unt } is relatively compact in L2 ((−r, 0); H01 (Ω)), and
0
(ii) gives that {unt } is relatively compact in L2 ((−r, 0); L2 (Ω)). Therefore, we
conclude that {U (t, τ )} is pullback ω-D-limit compact in X, which completes the
proof.
Theorem 3.10. The process {U (t, τ )} corresponding to (1.4) has a pullback Db = {A(t) : t ∈ R} in X.
attractor A
Proof. From Lemma 3.8, {U (t, τ )} has a family of Pullback D-absorbing sets in X.
By Theorem 2.8, it remains to show that {U (t, τ )} is Pullback w-D-limit compact.
To this end, we need to check conditions (1)-(3) in Proposition 3.9. To this aim,
we decompose f as
f = f0 + f1 ,
1
where f0 , f1 ∈ C (R, R) satisfy
f0 (u)u ≥ −c1 u2 − c2 ,
f00 (u)
≥ −c3 ,
(3.55)
(3.56)
α
(3.57)
α
(3.58)
|f0 (u)| ≤ k(1 + |u| ) ,
|f1 (u)| ≤ k(1 + |u| ) .
The delayed forcing term b is decomposed as
b = b0 + b1 ,
2
where b0 , b1 : R × L ((−r, 0); L (Ω)) → L2 (Ω) satisfy
(a) b0 (t, 0) = 0 for all t ∈ R;
(b) there exists Cb0 > 0 such that for all t ≥ τ and all u, v ∈ L2 ([τ −r, t]; L2 (Ω)),
Z t
Z t
kb0 (s, us ) − b0 (s, vs )k2 ds ≤ Cb0
ku(s) − v(s)k2 ds ;
(3.59)
τ
2
τ −r
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
19
(c) b1 (t, 0) = 0 for all t ∈ R;
(d) there exists Cb1 > 0 such that for all t ≥ τ and all u, v ∈ L2 ([τ −r, t]; L2 (Ω)),
Z t
Z t
eσs kb1 (s, us ) − b1 (s, vs )k2 ds ≤ Cb1
eσs ku(s) − v(s)k2 ds ,
(3.60)
τ −r
τ
Let λ1 , λ2 , . . . be the eigenvalues of −∆ in H01 (Ω) and w1 , w2 , . . . the corresponding
eigenfunctions. Then we have 0 < λ1 ≤ λ2 ≤ . . . ≤ λn → +∞ as n → +∞.
Then {w1 , w2 , . . .} form an orthogonal basis in L2 (Ω) and H01 (Ω). Let Vn =
span{w1 , w2 , . . . , wn }, P be the canonical projector on Vn and I be the identity.
Then we decompose U (t, τ )(u0 , ϕ) = (u(t), ut ) as
u(t) = P u(t) + (I − P )u(t) = v(t) + w(t),
and
ut = P ut + (I − P )ut = vt + wt .
Here v and w solve the following problems:
∂
∂
v − ∆ v − ∆v + f0 (v) = b0 (t, vt )
∂t
∂t
v = 0 on ∂Ω
(3.61)
v(τ, x) = P u0
v(τ + θ, x) = P ϕ(θ), θ ∈ (−r, 0)
and
∂
∂
w − ∆ w − ∆w + f (u) − f0 (v) = b(t, ut ) − b0 (t, vt ) + g
∂t
∂t
w = 0 on ∂Ω
(3.62)
w(τ, x) = (I − P )u0
w(τ + θ, x) = (I − P )ϕ(θ),
θ ∈ (−r, 0)
First, we establish that for all τ ≤ τ0 , (u(t), ut ) ∈ U (t, τ )B(τ ) and satisfies
kP (u(t), ut )kX1 < +∞. To do this, we multiply (3.61) by v and integrating over
Ω; to obtain
Z
Z
d
2
2
2
(kv(t)k + k∇v(t)k ) + 2k∇v(t)k + 2
f0 (v)v = 2
b0 (t, vt )v .
dt
Ω
Ω
By (3.55) and the Cauchy inequality one obtains
d
(kv(t)k2 + k∇v(t)k2 ) + 2k∇v(t)k2
dt
1
≤ 2c1 kv(t)k2 + 2c2 |Ω| + kb0 (t, vt )k2 + ε4 kv(t)k2
ε4
1
≤ (2c1 + ε4 )kv(t)k2 + 2c2 |Ω| + kb0 (t, vt )k2 .
ε4
Integrating from τ to t, for τ ≤ t ≤ T , we have
Z t
2
2
kv(t)k + k∇v(t)k + 2
k∇v(s)k2 ds
τ
Z t
2
2
≤ kv(τ )k + k∇v(τ )k + (2c1 + ε4 )
kv(s)k2 ds
τ
20
H. HARRAGA, M. YEBDRI
+
1
ε4
Z
EJDE-2016/07
t
kb0 (s, vs )k2 ds + 2c2 |Ω|(t − τ ) .
τ
Using (3.59) and (a), one has
2
Z
2
t
k∇v(s)k2 ds
Z t
2
2
≤ kv(τ )k + k∇v(τ )k + (2c1 + ε4 )
kv(s)k2 ds
τ
Z
Cb t
+
kv(s)k2 ds + 2c2 |Ω|(t − τ )
ε4 τ
Z t
2
2
kv(s)k2 ds
≤ kv(τ )k + k∇v(τ )k + (2c1 + ε4 )
τ
Z τ
Z t
Cb
Cb
+ 0
kv(s)k2 ds + 0
kv(s)k2 ds + 2c2 |Ω|(t − τ ) .
ε4 τ −r
ε4 τ
kv(t)k + k∇v(t)k + 2
τ
So, one has
kv(t)k2 + k∇v(t)k2 + 2
t
Z
k∇v(s)k2 ds
Z
Cb0 τ
2
2
kv(s)k2 ds
≤ kv(τ )k + k∇v(τ )k +
ε4 τ −r
Z t
Cb
+ (2c1 + ε4 + 0 )
kv(s)k2 ds + 2c2 |Ω|(t − τ ) .
ε4
τ
τ
By (3.3), one obtains
kv(t)k2 + k∇v(t)k2 + 2
Z
t
k∇v(s)k2 ds
τ
Z τ
Cb
2
2
kv(s)k2 ds
≤ kv(τ )k + k∇v(τ )k + 0
ε4 τ −r
C
Z t
2c1 + ε4 + εb40
k∇v(s)k2 ds + 2c2 |Ω|(t − τ ) .
+
λ1
τ
Thus, one finds that
2
2
kv(t)k + k∇v(t)k + 2 −
≤ kv(τ )k2 + k∇v(τ )k2 +
2c1 + ε4 +
λ1
Cb0
ε4
By the Theorem 3.2, we have η1 := 2 −
gives
k∇v(t)k2 ≤ kv(τ )k2 + k∇v(τ )k2 +
Cb
ε4
Z
Z
t
k∇v(s)k2 ds
τ
τ
kv(s)k2 ds + 2c2 |Ω|(t − τ ) .
τ −r
2c1 +ε4 +
λ1
Cb0
ε4
Z
Cb
0
ε4
> 0. So the previous estimate
τ
τ −r
kv(s)k2 ds + 2c2 |Ω|(T − τ ) ,
(3.63)
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
21
and
t
Z
k∇v(s)k2 ds
τ
Z τ
n
o
Cb
≤ η1−1 kv(τ )k2 + k∇v(τ )k2 + 0
kv(s)k2 ds + 2c2 |Ω|(T − τ ) .
ε4 τ −r
(3.64)
Therefore, for t − r ≥ τ , with([t − r, t] ⊂ [τ, t]), we have
Z t
Z t
k∇v(s)k2 ds ≤
k∇v(s)k2 ds
t−r
τ
≤ η1−1 (kv(τ )k2 + k∇v(τ )k2 +
Cb0
ε4
Z
τ
kv(s)k2 ds)
(3.65)
τ −r
+ 2c2 η1−1 |Ω|(T − τ ) .
We add (3.63) and (3.65) to obtain
Z t
2
k∇v(t)k +
k∇v(s)k2 ds
t−r
Z
n
o
Cb0 τ
(3.66)
−1
2
2
≤ (1 + η1 ) kv(τ )k + k∇v(τ )k +
kv(s)k2 ds + 2c2 |Ω|(T − τ )
ε4 τ −r
:= M 2 .
Hence, one obtains
kP (u(t), ut )kX1 ≤ M ,
which means that the condition (1) in Proposition 3.9 holds true.
Now, taking the inner product in L2 (Ω) of (3.62) with w, we obtain
Z
d
(kw(t)k2 + k∇w(t)k2 ) + 2k∇w(t)k2 + 2 (f (u) − f0 (v))w
dt
Ω
Z
Z
= 2 (b(t, ut ) − b0 (t, vt ))w + 2
g w.
Ω
(3.67)
Ω
Since f0 (v) = f (v) − f1 (v) and b0 (t, vt ) = b(t, vt ) − b1 (t, vt ), we obtain
Z
Z
d
(kw(t)k2 + k∇w(t)k2 ) + 2k∇w(t)k2 + 2
|f (u) − f (v)||w| + 2
|f1 (v)||w|
dt
Ω
Ω
Z
Z
Z
≤2
|b(t, ut ) − b(t, vt )| |w| + 2
|b1 (t, vt )||w| + 2
|g| |w| .
Ω
Ω
Ω
By (2.2), we have
Z
(f (u) − f (v))(u − v) ≥ −c3 ku − vk2 .
(3.68)
Ω
Thus, by (3.68) and the Cauchy inequality, (3.67) leads to
d
(kw(t)k2 + k∇w(t)k2 ) + 2k∇w(t)k2
dt
≤ (2c3 + ε1 + ε2 + ε5 )kw(t)k2
(3.69)
Z
1
1
1
+2
|f1 (v)| |w| + kb(t, ut ) − b(t, vt )k2 + kb1 (t, vt )k2 + kg(t)k2 .
ε1
ε5
ε2
Ω
22
H. HARRAGA, M. YEBDRI
EJDE-2016/07
By (3.58) and the Holder inequality,
Z
Z
|f1 (v)| |w| ≤ k (1 + |v|α )|w|
Ω
Ω
≤k
Z
α
(1 + |v| )
+2 Z
N2N
2N
N +2
≤c
−2
N2N
Ω
Ω
Z
2N
|w| N −2
(1 + |v|
α N2N
+2
)
+2
N2N
kwk
2N
L N −2 (Ω)
Ω
.
4
2N
2N
N +2
Since α < min{ N
−2 , 2 + N }, one has α N +2 < N −2 for all N ≥ 3. Then
Z
+2
Z
N2N
2N
|f1 (v)| |w| ≤ c
kwk N2N
(1 + c|v| N −2 )
−2
Ω
L
Ω
2N
N −2
2N
L N −2
+2
N2N
(Ω)
kwk
2N
As above, one gets
Z
2N N +2
N +2
2N
|f1 (v)| |w| ≤ c(|Ω| 2N + ckvk N −2
)kwk
2N
2N
≤ c |Ω| + ckvk
L N −2 (Ω)
(Ω)
L N −2 (Ω)
Ω
.
L N −2 (Ω)
N +2
≤ c(1 + kvk N −2
2N
L N −2 (Ω)
)kwk
2N
L N −2 (Ω)
.
2N
By the embedding of H01 (Ω) in L N −2 (Ω), we have
Z
N +2
N +2
|f1 (v)| |w| ≤ c(1 + ck∇vk N −2 )k∇wk ≤ c(1 + k∇vk N −2 )k∇wk
Ω
By (3.66), we obtain
Z
N +2
|f1 (v)| |w| ≤ c(1 + M N −2 )k∇wk ≤ ck∇wk ,
Ω
as k∇vk
N +2
N −2
N +2
N −2
≤M
, and
Z
c2
ε3
ε3
|f1 (v)| |w| ≤
+ k∇w(t)k2 ≤ c + k∇w(t)k2 ,
2ε
2
2
3
Ω
via the Cauchy inequality. By the above estimate and (3.69), one obtains
d
(kw(t)k2 + k∇w(t)k2 ) + (2 − ε3 )k∇w(t)k2
dt
1
≤ (2c3 + ε1 + ε2 + ε5 )kw(t)k2 + kb(t, ut ) − b(t, vt )k2
ε1
1
1
2
2
+ kb1 (t, vt )k + kg(t)k + c .
ε5
ε2
Using (3.3), one has
d
(kw(t)k2 + k∇w(t)k2 ) + (2 − ε3 )k∇w(t)k2
dt
2c3 + ε1 + ε2 + ε5
1
1
≤
k∇w(t)k2 + kb(t, ut ) − b(t, vt )k2 + kb1 (t, vt )k2
λ1
ε1
ε5
1
+ kg(t)k2 + c .
ε2
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
23
For λ1 > c3 , ε3 < 1, and ε1 , ε2 , ε5 small enough, we have
2 − ε3 −
2c3 + ε1 + ε2 + ε5
> 0.
λ1
So one gets
d
2c3 + ε1 + ε2 + ε5
(kw(t)k2 + k∇w(t)k2 ) + (2 − ε3 −
)k∇w(t)k2
dt
λ1
1
1
1
≤ kb(t, ut ) − b(t, vt )k2 + kb1 (t, vt )k2 + kg(t)k2 + c .
ε1
ε5
ε2
1
Similarly as in (3.27), we can choose the positive constant δ 0 < min{2c4 , 2λ2λ
}
1 +1
such that
2c3 + ε1 + ε2 + ε5 k∇w(t)k2 .
δ 0 (k∇w(t)k2 + kw(t)k2 ) ≤ 2 − ε3 −
λ1
In fact,
1
1
1
d
y(t) + δ 0 y(t) ≤ kb(t, ut ) − b(t, vt )k2 + kb1 (t, vt )k2 + kg(t)k2 + c ,
dt
ε1
ε5
ε2
where y(t) = k∇w(t)k2 + kw(t)k2 . Multiplying this last inequality by eσt , such that
σ < δ 0 , to find that
d
y(t) + δ 0 eσt y(t)
dt
1
1
1
≤ eσt kb(t, ut ) − b(t, vt )k2 + eσt kb1 (t, vt )k2 + eσt kg(t)k2 + ceσt .
ε1
ε5
ε2
eσt
On the other hand, we have
d σt
d
(e y(t)) = σeσt y(t) + eσt y(t)
dt
dt
≤ (σ − δ 0 )eσt y(t) + eσt
+
1
1
kb(t, ut ) − b(t, vt )k2 + eσt kb1 (t, vt )k2
ε1
ε5
1 σt
e kg(t)k2 + ceσt .
ε2
Integrating from τ to t, we obtain
Z t
y(t) ≤ e−σ(t−τ ) y(τ ) + (σ − δ 0 )e−σt
eσs y(s)ds
τ
Z t
Z t
1
1
eσs kb(s, us ) − b(s, vs )k2 ds + e−σt
eσs kb1 (s, vs )k2 ds
+ e−σt
ε1
ε
5
τ
τ
Z
Z t
1 −σt t σs
2
−σt
σs
+ e
e kg(s)k ds + ce
e ds
ε2
τ
τ
Z t
≤ e−σ(t−τ ) y(τ ) + (σ − δ 0 )e−σt
eσs y(s)ds
τ
Z
Z
1 −σt t σs
1 −σt t σs
2
+ e
e kb(s, us ) − b(s, vs )k ds + e
e kb1 (s, vs )k2 ds
ε1
ε5
τ
τ
Z
1 −σt t σs
+ e
e kg(s)k2 ds + c(1 − e−σ(t−τ ) ) .
ε2
τ
24
H. HARRAGA, M. YEBDRI
EJDE-2016/07
We use (3.23), (3.60) and (c) to get
Z t
y(t) ≤ e−σ(t−τ ) y(τ ) + (σ − δ 0 )e−σt
eσs y(s)ds
τ
Z
Z t
Cb −σt t σs
Cb
+
e ku(s) − v(s)k2 ds + 0 e−σt
eσs kv(s)k2 ds
e
ε1
ε5
τ −r
τ −r
Z t
1
+ e−σt
eσs kg(s)k2 ds + c(1 − e−σ(t−τ ) )
ε2
τ
Z t
−σ(t−τ )
eσs y(s)ds
≤e
y(τ ) + (σ − δ 0 )e−σt
τ
Z
Z
Cb −σt τ σs
Cb −σt t σs
2
+
e
e kw(s)k ds +
e
e kw(s)k2 ds
ε1
ε1
τ −r
τ
Z
Z
Cb0 −σt t σs
Cb0 −σt τ σs
2
e
e
e kv(s)k ds +
e kv(s)k2 ds
+
ε5
ε5
τ −r
τ
Z t
1
+ e−σt
eσs kg(s)k2 ds + c .
ε2
τ
Since
Z
t
σs
2
e kw(s)k ds ≤ e
τ −r
Z t
στ
eσs kv(s)k2 ds ≤ eστ
τ −r
τ
Z
2
Z
t
kw(s)k ds +
τ −r
Z τ
eσs kw(s)k2 ds ,
τ
kv(s)k2 ds + eσt
τ −r
Z
t
kv(s)k2 ds ,
τ
by (3.3) and (3.64), one obtains
−σ(t−τ )
0
−σt
Z
t
y(τ ) + (σ − δ )e
eσs y(s)ds
τ
Z
Z
Cb −σt t σs
Cb −σ(t−τ ) τ
e
e
kw(s)k2 ds +
e k∇w(s)k2 ds
+
ε1
ε1 λ1
τ −r
τ
Z t
Z τ
Cb
Cb0
+ 0 e−σ(t−τ )
k∇v(s)k2 ds
kv(s)k2 ds +
ε5
ε
λ
5
1
τ
τ −r
Z
1 −σt t σs
+ e
e kg(s)k2 ds + c
ε2
τ
Z t
−σ(t−τ )
0 −σt
≤e
y(τ ) + (σ − δ )e
eσs y(s)ds
τ
Z
Z
Cb −σ(t−τ ) τ
Cb −σt t σs
2
+
e
kw(s)k ds +
e
e k∇w(s)k2 ds
ε1
ε1 λ1
τ −r
τ
Z τ
Z t
Cb
1
+ 0 e−σ(t−τ )
kv(s)k2 ds + e−σt
eσs kg(s)k2 ds + c
ε5
ε2
τ −r
τ
Z τ
o
Cb0 −1 n
Cb
+
η1 kv(τ )k2 + k∇v(τ )k2 + 0
kv(s)k2 ds + 2c2 |Ω|(T − τ ) .
ε5 λ1
ε4 τ −r
y(t) ≤ e
For µ0 := δ 0 − σ −
Cb
ε1 λ1
> 0, one has
Z t
2
2
0 −σt
k∇w(t)k + kw(t)k + µ e
eσs k∇w(s)k2 ds
τ
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
25
Z
Z τ
Cb τ
Cb
kw(s)k2 ds + 0
kv(s)k2 ds)
ε1 τ −r
ε5 τ −r
Z τ
o
Cb0 −1 n
Cb
+
kv(s)k2 ds + 2c2 |Ω|(T − τ )
η1 kv(τ )k2 + k∇v(τ )k2 + 0
ε5 λ1
ε4 τ −r
Z t
1
eσs kg(s)k2 ds + c ,
(3.70)
+ e−σt
ε2
τ
≤ e−σ(t−τ ) (k∇w(τ )k2 + kw(τ )k2 +
whereupon
k∇w(t)k2 + kw(t)k2
Z
Z
Cb τ
Cb0 τ
2
2
2
−σ(t−τ )
k∇w(τ )k + kw(τ )k +
kw(s)k ds +
kv(s)k2 ds
≤e
ε1 τ −r
ε5 τ −r
Z τ
n
o
Cb0 −1
Cb
+
η1 kv(τ )k2 + k∇v(τ )k2 + 0
kv(s)k2 ds + 2c2 |Ω|(T − τ )
ε5 λ 1
ε4 τ −r
Z t
1
+ e−σt
eσs kg(s)k2 ds + c ,
ε2
τ
(3.71)
and
Z t
µ0 e−σt
eσs k∇w(s)k2 ds
τ
Z
Z τ
Cb
Cb τ
−σ(t−τ )
kw(s)k2 ds + 0
kv(s)k2 ds)
≤e
(k∇w(τ )k2 + kw(τ )k2 +
ε1 τ −r
ε5 τ −r
Z τ
o
Cb0 −1 n
Cb
+
η1 kv(τ )k2 + k∇v(τ )k2 + 0
kv(s)k2 ds + 2c2 |Ω|(T − τ )
ε5 λ 1
ε4 τ −r
Z t
1
+ e−σt
eσs kg(s)k2 ds + c ,
ε2
τ
(3.72)
For t − r ≥ τ , we have
t
Z
eσs k∇w(s)k2 ds ≥
τ
Z
t
eσs k∇w(s)k2 ds ≥ eσ(t−r)
t−r
Z
t
k∇w(s)k2 ds .
t−r
So, by this inequality, (3.72) becomes
Z
t
k∇w(s)k2 ds
t−r
Z
Cb τ
kw(s)k2 ds
≤ µ0−1 e−σ(t−τ −r) k∇w(τ )k2 + kw(τ )k2 +
ε1 τ −r
Z τ
µ0−1 η −1 C
n
Cb
b0 σr
1
+ 0
kv(s)k2 ds +
e
kv(τ )k2
ε5 τ −r
ε5 λ1
Z τ
o
Cb
kv(s)k2 ds + 2c2 |Ω|(T − τ )
+ k∇v(τ )k2 + 0
ε4 τ −r
Z t
0−1
µ
−σ(t−r)
e
eσs kg(s)k2 ds + ceσr .
+
ε2
τ
(3.73)
26
H. HARRAGA, M. YEBDRI
EJDE-2016/07
We add (3.71) and (3.73), and we use (3.24) to obtain
Z t
k∇w(s)k2 ds
k∇w(t)k2 +
t−r
≤ k∇w(t)k2 + kw(t)k2 +
Z
t
k∇w(s)k2 ds
t−r
Z
n
Cb τ
2
−σ(t−τ )
k∇w(τ )k + kw(τ )k2 +
≤ e
kw(s)k2 ds
ε1 τ −r
Z
η −1 C n
Cb0 τ
b0
+
kv(s)k2 ds + 1
kv(τ )k2
ε5 τ −r
ε5 λ1
Z
o
Cb0 τ
2
kv(s)k2 ds + 2c2 |Ω|(T − τ )
+ k∇v(τ )k +
ε4 τ −r
Z t
o
1
+ ( e−σt
eσs kg(s)k2 ds + c) (1 + µ0−1 eσr ) .
ε2
−∞
(3.74)
Then, (3.74) shows that for all ε > 0 , τ ≤ τ0 and all (u(t), ut ) ∈ U (t, τ )B(τ ), one
has
k(I − P )(u(t), ut )k2X1 ≤ ε2 .
Finally, by considering the ordinary functional differential system (3.61), we have
k∆vk2 ≤ λm k∇vk2 ≤ λ2m kvk2 ,
as
m
m
m
X
X
X
k∇vk2 = h∆v, vi = h hv, wi iλi wi ,
hv, wj iwj i =
λi hv, wi i2 ,
i=1
and
(3.75)
j=1
i=1
k∆vk2 = h∆v, ∆vi
m
m
X
X
hv, wj iλj wj i
= h hv, wi iλi wi ,
=
j=1
i=1
m
X
λ2i hv, wi i2 ≤ λm
i=1
m
X
(3.76)
λi hv, wi i2 .
i=1
Now, we check the equicontinuity property of the solutions {v(·)} in the space
L2 ([t − r, t]; L2 (Ω)). Then, for any t1 ∈ [t − r, t], any l ∈ R+ with l < δ and for
[t1 , t1 + l] ⊂ [t − r, t], we have
kTl v(t1 ) − v(t1 )k = kv(t1 + l) − v(t1 )k
Z 1
dv(t1 + s)
≤
k
kds
dt1
0
Z 1
Z 1
d
≤
k∆
v(t1 + s)kds +
k∆v(t1 + s)kds
dt1
0
0
Z 1
Z 1
+
kf0 (v)kds +
kb0 (t1 + s, vt1 +s )kds ,
0
0
and so
2
kv(t1 + l) − v(t1 )k ≤
Z
0
1
d
k∆
v(t1 + s)kds +
dt1
Z
1
k∆v(t1 + s)kds
0
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
Z
1
Z
kf0 (v)kds +
+
0
1
27
2
kb0 (t1 + s, vt1 +s )kds .
0
Consequently,
kv(t1 + l) − v(t1 )k2
Z 1
Z 1
2
d
≤2
k∆
k∆v(t1 + s)kds
v(t1 + s)kds +
dt1
0
0
Z 1
Z 1
2
+2
kf0 (v)kds +
kb0 (t1 + s, vt1 +s )kds
0
0
1
2
Z 1
2
d
≤4
k∆
v(t1 + s)kds + 4
k∆v(t1 + s)kds
dt1
0
0
Z 1
2
Z 1
2
+4
kf0 (v)kds + 4
kb0 (t1 + s, vt1 +s )kds
Z
0
0
By Holder inequality,
kv(t1 + l) − v(t1 )k2
Z 1
Z 1
d
2
v(t1 + s)k ds + 4l
k∆v(t1 + s)k2 ds
≤ 4l
k∆
dt1
0
0
Z 1
Z 1
2
+ 4l
kf0 (v)k ds + 4l
kb0 (t1 + s, vt1 +s )k2 ds
Z
0
1
0
d
≤ 4l
k∆
v(t1 + s)k2 ds +
dt1
0
Z 1
+
kb0 (t1 + s, vt1 +s )k2 ds ,
1
Z
k∆v(t1 + s)k2 ds +
Z
0
1
kf0 (v)k2 ds
0
0
which after integration over [t − r, t] leads to
Z t
kv(t1 + l) − v(t1 )k2 dt1
t−r
Z 1
d
2
≤ 4l
v(t1 + s)k ds +
k∆
k∆v(t1 + s)k2 ds
dt1
t−r
0
0
Z 1
Z 1
+
kf0 (v)k2 ds +
kb0 (t1 + s, vt1 +s )k2 ds dt1
Z
t
Z
1
0
0
Z 1Z t
d
v(t1 + s)k2 dt1 ds +
k∆v(t1 + s)k2 dt1 ds
dt
1
0
t−r
0
t−r
Z 1Z t
Z 1Z t
+
kf0 (v)k2 dt1 ds +
kb0 (t1 + s, vt1 +s )k2 dt1 ds .
≤ 4l
Z
0
1
Z
t
k∆
t−r
0
t−r
Next we will estimate the five terms on the right-hand side of the equation.
By (3.75), we have
Z t
Z t
k∆v(t1 + s)k2 dt1 ≤ λm
k∇v(t1 + s)k2 dt1 .
t−r
t−r
From (3.25), one has
Z t
Z
2N
N −2
k∇v(t1 + s)k2 dt1 ≤ ck(u0 , ϕ)kX
t−r
t
t−r
e−σ(t1 +s−τ ) dt1
(3.77)
28
H. HARRAGA, M. YEBDRI
Z
t
+c
EJDE-2016/07
Z
1 + e−σ(t1 +s)
t1 +s
0
eσs kg(s0 )k2 ds0 dt1
−∞
t−r
2N c
N −2
≤ k(u0 , ϕ)kX
e−σ(t−r+s−τ ) − e−σ(t+s−τ ) + cr
σ
Z t
0
c −σ(t−r+s)
−σ(t+s)
eσs kg(s0 )k2 ds0
+
e
−e
σ
−∞
2N
c
N −2 −σ(t−r+s−τ )
0
≤ k(u , ϕ)kX e
σ
Z
c −σ(t−r+s) t σs0
e kg(s0 )k2 ds0 + cr .
(3.78)
+ e
σ
−∞
Integrating over [0, l], we obtain
Z 1Z t
k∇v(t1 + s)k2 dt1 ds
t−r
0
2N
c
N −2 −σ(t−r−τ )
≤ 2 k(u0 , ϕ)kX
e
(1 − e−σl )
σ
Z t
c
+ 2 e−σ(t−r) (1 − e−σl )
eσs kg(s)k2 ds + crl .
σ
−∞
By (3.75) and (3.79), we have
Z 1Z t
2N
c
N −2 −σ(t−r−τ )
e
(1 − e−σl )
k∆v(t1 + s)k2 dt1 ds ≤ λm 2 k(u0 , ϕ)kX
σ
0
t−r
Z t
c
+ λm 2 e−σ(t−r) (1 − e−σl )
eσs kg(s)k2 ds + λm crl → 0 as l → 0 .
σ
−∞
From (a) and (3.59), we obtain
Z t
Z
kb0 (t1 + s, vt1 +s )k2 dt1 ≤ Cb0
t−r
(3.79)
(3.80)
t
kv(t1 + s)k2 dt1 .
t−2r
Using (3.3), one has
Z t
Z
−1
2
kb0 (t1 + s, vt1 +s )k dt1 ≤ Cb0 λ1
t−r
t
k∇v(t1 + s)k2 dt1 .
t−2r
By (3.25), one gets
Z t
kb0 (t1 + s, vt1 +s )k2 dt1
t−r
2N
N −2
0
≤ Cb0 λ−1
1 ck(u , ϕ)kX
+ Cb0 λ−1
1 c
Z
Z
t
e−σ(t1 +s−τ ) dt1
t−2r
t
t−2r
dt1 + Cb0 λ−1
1 c
Z
t
t−2r
e−σ(t1 +s)
Z
t1 +s
0
eσs kg(s0 )k2 ds0 dt1
−∞
2N
c
N −2
≤
k(u0 , ϕ)kX
(e−σ(t−2r+s−τ ) − e−σ(t+s−τ ) )
σ
Z t
0
−1
−1 c
−σ(t−2r+s)
−σ(t+s)
+ 2Cb0 λ1 cr + Cb0 λ1
e
−e
eσs kg(s0 )k2 ds0
σ
−∞
2N
−1 c
N −2 −σ(t−2r+s−τ )
0
≤ Cb0 λ1
k(u , ϕ)kX e
σ
Cb0 λ−1
1
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
−1 c −σ(t−2r+s)
+ 2Cb0 λ−1
e
1 cr + Cb0 λ1
σ
Z
t
29
0
eσs kg(s0 )k2 ds0 .
(3.81)
−∞
Integrating over [0, l], we obtain
Z 1Z t
kb0 (t1 + s, vt1 +s )k2 dt1 ds
t−r
0
Z 1
2N
c
N −2
0
≤
k(u , ϕ)kX
e−σ(t−2r+s−τ ) ds
σ
0
Z t
Z 1
Z 1
−1
−1 c
σs0
0 2 0
−σ(t−2r+s)
e kg(s )k ds ds + 2Cb0 λ1 cr
ds
+ Cb0 λ1
e
σ 0
−∞
0
2N
c
N −2 −σ(t−2r−τ )
≤ Cb0 λ−1
k(u0 , ϕ)kX
e
(1 − e−σl )
1
2
σ
Z t
0
c −σ(t−2r)
−σl
eσs kg(s0 )k2 ds0 + 2Cb0 λ−1
e
(1
−
e
)
(3.82)
+ Cb0 λ−1
1 crl
1
2
σ
−∞
→ 0 as l → 0 .
Cb0 λ−1
1
Now, it is clear from (3.57) that
kf0 (v)k2 ≤
Z
k 2 (1 + |v|α )2 dx .
Ω
By (3.3) and the fact that 2α < N4N
−2 , one has
Z
kf0 (v)k2 ≤
2k 2 (1 + |v|2α )dx
Ω
2
≤ 2k |Ω| + 2k 2 kv(t1 + s)k2α
4N
N −2 .
≤ 2k 2 |Ω| + 2k 2 µ̃λ−1
1 k∇v(t1 + s)k
(3.83)
By (3.25), we have
n
2N
4N
N −2
k∇v(t1 + s)k N −2 ≤ ce−σ(t1 +s−τ ) k(u0 , ϕ)kX
Z t1 +s
o N2N
0
−2
.
+ c(1 + e−σ(t1 +s)
eσ(s ) kg(s0 )k2 ds0 )
−∞
Similarly, one obtains
4N
k∇v(t1 + s)k N −2
2N 2N
2N
N −2
N −2
≤ 2 N −2 −1 ce−σ(t1 +s−τ ) k(u0 , ϕ)kX
Z t1 +s
N2N
0
2N
2N
−2
eσs kg(s0 )k2 ds0
+ 2 N −2 −1 c N −2 1 + e−σ(t1 +s)
−∞
N +2
N −2
2N
N −2
(
−σ(t1 +s−τ ) N2N
−2
2N
)2
2N +4
2N
k(u0 , ϕ)kXN −2 + 2 N −2 c N −2
Z t1 +s
N2N
2N +4
0
2N
−2
−σ(t1 +s) N2N
N
−2
N
−2
−2
+2
c
e
eσs kg(s0 )k2 ds0
.
≤2
c
e
−∞
Hence by (3.83) and (3.84), one obtains
Z t
kf0 (v)k2 dt1
t−r
(3.84)
30
H. HARRAGA, M. YEBDRI
≤ 2k 2 |Ω|
Z
t
1dt1 + 2k 2 µ̃λ−1
1
Z
t−r
t
EJDE-2016/07
4N
k∇v(t1 + s)k N −2 dt1
t−r
N +2
2N
2N
2N k2
( 2N )2
N −2 c N −2
− e−σ(t+s−τ ) N −2 + e−σ(t−r+s−τ ) N −2 k(u0 , ϕ)kXN −2
≤ 2 µ̃λ−1
1 2
σ
N +2
2N
2N
2N k2
N −2 c N −2 e−σ(t−r+s) N −2 − e−σ(t+s) N −2
+ 2 µ̃λ−1
1 2
σ
N2N
Z t
3N +2
0
2N
−2
×
eσs kg(s0 )k2 ds0
+ 2k 2 |Ω|r + 2 N −2 c N −2 k 2 µ̃λ−1
1 r
−∞
N +2
2N
2N
k2
( 2N )2
N −2 c N −2 e−σ(t−r+s−τ ) N −2 k(u0 , ϕ)k N −2
≤ 2 µ̃λ−1
1 2
X
σ
Z t
2N
N +2
2N
2N
k2
σs0
0 2 0 N −2
N −2 c N −2 e−σ(t−r+s) N −2
2
e
+ 2 µ̃λ−1
kg(s
)k
ds
1
σ
−∞
3N +2
2N
+ 2k 2 |Ω|r + 2 N −2 c N −2 k 2 µ̃λ−1
1 r,
which integrated from 0 to l gives
Z 1Z t
kf0 (v)k2 dt1 ds
0
t−r
2
Z 1
N +2
2N
2N
k
( 2N )2
N −2 c N −2 k(u0 , ϕ)k N −2
µ̃λ−1
2
e−σ(t−r+s−τ ) N −2 ds
1
X
σ
0
Z t
2N Z 1
N +2
0
2N
2N
k 2 −1 N
σs
0
2 0 N −2
−2
N
−2
+ 2 µ̃λ1 2
c
e kg(s )k ds
e−σ(t−r+s) N −2 ds
σ
−∞
0
Z 1
3N +2
2N
+
(2k 2 |Ω|r + 2 N −2 c N −2 k 2 µ̃λ−1
1 r) .ds
≤2
0
Then, we find
Z 1Z t
kf0 (v)k2 dt1 ds
0
t−r
N +2
2N
2N
k2
( 2N )2 −σ(t−r−τ ) 2N
N −2 c N −2 k(u0 , ϕ)k N −2
N −2 (1 − e−σl N −2 )
2
≤ 2 2 µ̃λ−1
e
1
X
σ
Z t
N2N
N +2
0
2N
2N
k2
−2 −σ(t−r) 2N
N −2 c N −2
N −2 (1 − e−σl N −2 )
2
e
+ 2 2 µ̃λ−1
eσs kg(s0 )k2 ds0
1
σ
−∞
3N +2
2N
+ 2k 2 |Ω|rl + 2 N −2 c N −2 k 2 µ̃λ−1
1 rl → 0
as l → 0 .
(3.85)
R1Rt
It remains to estimate 0 t−r k∆ dtd1 v(t1 + s)k2 dt1 ds. To this end, we take the
inner product in L2 (Ω) of (3.61) with −∆ ∂t∂1 v, we find
d
1 d
d
k∇
v(t1 + s)k2 + k∆
v(t1 + s)k2 +
k∆v(t1 + s)k2
dt1
dt1
2 dt1
Z
Z
∂
∂
≤
f0 (v)∆
v+
b0 (t1 + s, vt1 +s )(−∆
v) .
∂t1
∂t1
Ω
Ω
We have
Z
∂
v=−
f0 (v)∆
∂t
1
Ω
Z
Ω
f00 (v)∇v∇
∂
v.
∂t1
(3.86)
EJDE-2016/07
DIFFUSION EQUATIONS WITH DELAY
By (3.56), it follows that
Z
Z
∂ ∂ ∂
f0 (v)∆
v ≤ c3
v ≤ c3 k∇vk∇
v .
|∇v|∇
∂t1
∂t1
∂t1
Ω
Ω
Using the Young inequality, one has
Z
∂
c2
1
d
f0 (v)∆
v ≤ 3 k∇v(t1 + s)k2 + k∇
v(t1 + s)k2 .
∂t
2
2
dt
1
1
Ω
31
(3.87)
By (3.87) and (3.86), one finds
d
d
1 d
k∇
v(t1 + s)k2 + k∆
v(t1 + s)k2 +
k∆v(t1 + s)k2
dt1
dt1
2 dt1
Z
c23
1
d
∂
2
2
≤ k∇v(t1 + s)k + k∇
v(t1 + s)k +
v) .
b0 (t1 + s, vt1 +s )(−∆
2
2
dt1
∂t1
Ω
Which via the Cauchy inequality gives
d
1 d
d
v(t1 + s)k2 + k∆
v(t1 + s)k2 +
k∆v(t1 + s)k2
k∇
dt1
dt1
2 dt1
1
d
c2
1
≤ 3 k∇v(t1 + s)k2 + k∇
v(t1 + s)k2 +
kb0 (t1 + s, vt1 +s )k2
2
2
dt1
2ν1
d
ν1
v(t1 + s)k2 .
+ k∆
2
dt1
So, one has
d
k∆
v(t1 + s)k2
dt1
d
d
1 d
1
v(t1 + s)k2 + k∆
v(t1 + s)k2 +
k∆v(t1 + s)k2
≤ k∇
2
dt1
dt1
2 dt1
1
d
ν1
c2
kb0 (t1 + s, vt1 +s )k2 + k∆
v(t1 + s)k2 .
≤ 3 k∇v(t1 + s)k2 +
2
2ν1
2
dt1
Therefore, one gets
d
1
ν1
c2
)k∆
v(t1 + s)k2 ≤ 3 k∇v(t1 + s)k2 +
kb0 (t1 + s, vt1 +s )k2 .
2
dt1
2
2ν1
For ν1 small enough, we have ν2 := 1 − ν21 > 0. So we can write
(1 −
d
c2
1
v(t1 + s)k2 ≤ 3 k∇v(t1 + s)k2 +
kb0 (t1 + s, vt1 +s )k2 .
dt1
2ν2
2ν1 ν2
Which integrated over [t − r, t] leads to
Z t
d
k∆
v(t1 + s)k2 dt1
dt
1
t−r
Z t
Z t
c23
1
2
k∇v(t1 + s)k dt1 +
kb0 (t1 + s, vt1 +s )k2 dt1 ;
≤
2ν2 t−r
2ν1 ν2 t−r
k∆
integrating the above inequality over [0, l], one obtains
Z 1Z t
d
k∆
v(t1 + s)k2 dt1 ds
dt1
0
t−r
Z 1Z t
Z 1Z t
c2
1
≤ 3
k∇v(t1 + s)k2 dt1 ds +
kb0 (t1 + s, vt1 +s )k2 dt1 ds .
2ν2 0 t−r
2ν1 ν2 0 t−r
32
H. HARRAGA, M. YEBDRI
EJDE-2016/07
By (3.79) and (3.82), it follows that
Z 1Z t
d
k∆
v(t1 + s)k2 dt1 ds
dt1
t−r
0
≤
2N
c23 c
N −2 −σ(t−r−τ )
k(u0 , ϕ)kX
e
(1 − e−σl )
2
2ν3 σ
Z t
c2 c
c2
+ 3 2 e−σ(t−r) (1 − e−σl )
eσs kg(s)k2 ds + 3 crl
2ν2 σ
2ν2
−∞
2N
1
c
N −2 −σ(t−2r−τ )
+
Cb λ−1 k(u0 , ϕ)kX
e
(1 − e−σl )
2ν1 ν2 0 1 σ 2
Z t
1
1
−1 c −σ(t−2r)
−σl
(1 − e )
eσs kg(s)k2 ds +
Cb0 λ1 2 e
Cb λ−1 crl .
+
2ν1 ν2
σ
ν1 ν2 0 1
−∞
Thus, we obtain
Z 1Z t
d
k∆
v(t1 + s)k2 dt1 ds
dt
1
0
t−r
2N
c23
1
N −2 −σ(t−r−τ )
0
σr c
e
(1 − e−σl )
+
Cb0 λ−1
1 e ) 2 k(u , ϕ)kX
2ν2
2ν1 ν2
σ
Z t
c23
1
−1 σr c −σ(t−r)
−σl
+(
(1 − e )
eσs kg(s)k2 ds
+
Cb λ e ) 2 e
2ν2
2ν1 ν2 0 1
σ
−∞
≤(
+
c23
1
crl +
Cb λ−1
1 crl → 0
2ν2
ν1 ν2
(3.88)
as l → 0 .
Comprehensively, from (3.77), (3.80), (3.82), (3.85) and (3.88), we have
Z t
kv(t1 + l) − v(t1 )k2 dt1 → 0 as l → 0 ,
t−r
which implies the needed equicontinuity. This shows that the condition (3) in
Proposition 3.9 holds, and thus the process on X is pullback w-D-limit compact.
Then from Lemma 3.8 and Theorem 2.8, we conclude the existence of a pullback
D-attractor which completes the proof.
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Hafidha Harraga
Dept of Mathematics, University Aboubekr Belkaid ot Tlemcen 13000, Tlemcen, Algeria
E-mail address: h.harraga@yahoo.fr
Mustapha Yebdri
Dept of Mathematics, University Aboubekr Belkaid ot Tlemcen 13000, Tlemcen, Algeria
E-mail address: yebdri@yahoo.com