R E S E A R C H
Open Access
Pullback attractors for a class of
non-autonomous reaction-diffusion
equations in
R
n
Qiangheng Zhang
**Correspondence:
[email protected] School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China
Abstract
The aim of this paper is to consider the dynamical behaviour for a class of
non-autonomous reaction-diffusion equations inRn, where the external forceg(x,t)
satisfies only a certain integrability condition. The existence of
(L2(Rn),L2(Rn))-D-pullback attractors and (L2(Rn),Lp(Rn))-D-pullback attractors is
obtained for this evolution equation.
MSC: Primary 35B41; secondary 35B45; 35K15
Keywords: pullback attractors; dynamical behaviour; non-autonomous equations
1 Introduction
In this paper, we consider the asymptotic behaviour of solutions for the following non-autonomous reaction-diffusion equations defined in the whole space:
⎧ ⎨ ⎩
ut–νu+λu+f(u) +a(x)f(u) =g(x,t), inRn×[τ,∞),
u(x,τ) =uτ, inRn,
(.)
whereνandλare positive constants. Assume that nonlinear termsf(u),f(u)∈C(R;R)
satisfy the following conditions:
α|u|p–β|u|≤f(u)u≤α|u|p+β|u| and f(u)≥–l (.)
withp> andλ>β,
α|u|p–β≤f(u)u≤α|u|p+β and f(u)≥–l (.)
withp> , whereαi,βi,i= , , , , andli,i= , are positive constants. Furthermore,a(x)
is a function inRnand the external forceg(x,t)∈L
loc(R;L(Rn)) satisfies the following
conditions:
a(x)∈LRn∩L∞Rn and a(x) > , (.) t
–∞
eσsg(x,s)
L(Rn)ds<∞, for allt∈R,σ∈(,λ–β). (.)
In the last decade, the autonomous and non-autonomous infinite dimensional dynam-ical systems have been studied extensively by many authors (see, e.g. [–] and the refer-ences therein). The concept of pullback attractors was proposed in [] when the authors considered the asymptotic behaviour of random dynamical systems. Such attractors is a parameterised family{A(t)}t∈Rof invariant compact sets, which attract the trajectories of the systems when the initial instant of time goes to –∞and the final time remains fixed. Later on, the pullback attractors were extended to non-autonomous dynamical systems. In the last two decades, the theory of pullback attractors has been developed for non-autonomous dynamical systems and random dynamical systems (see, e.g. [–] and the references therein). In [], the authors introduced the notion ofD-pullback attractors, which requires that the processU(t,τ) associated with the systems beD-pullback asymp-totically compact.
It is well known that the Sobolev embeddings are no longer compact in unbounded domain, and so it is difficult to verify the processU(t,τ) associated with the systems to be pullback asymptotically compact. To overcome this drawback, in [], using the idea of Wang [], the authors proved the existence of pullback attractors inL(Rn) andH(Rn) for
non-autonomous reaction-diffusion equations defined onRn. Recently, motivated by [],
the authors of [] gave a new method to prove the existence ofD-pullback attractors by using the technique of non-compactness measure, and this method only needs the process
U(t,τ) associated with the systems to be norm-to-weak continuous (see Definition .) in the phase space.
As we know, the solutions may be unbounded for many non-autonomous systems when time tends to infinity, and we cannot obtain the existence of a uniform attractor for these systems. So we prove the existence of a pullback attractor to overcome this drawback. In this paper, we use a different approach from the article [] to prove the existence of pull-back attractors, and we improve the model equation as Eq. (.), which amounts to putting a weight function partially on the nonlinearity. We can also replace the conditions for the nonlinearityf(u) as given in [] thatf(u) satisfies only a Sobolev growth rate with some weak assumptions. For Eq. (.), the (L(Rn),L(Rn))-global attractor, (L(Rn),Lp(Rn
))-global attractor and (L(Rn),H(Rn))-global attractor were proved in [, ]. Using the
new method in [], we prove the existence ofD-pullback attractors inL(Rn) for Eq. (.)
and, motivated by the idea in [, ], we obtain the existence ofD-pullback attractors in
Lp(Rn) for Eq. (.). This new method has been used successfully in many papers (see, e.g.
[, , , ] and the references therein).
For convenience, the letterCdenotes a constant which may be different from line to line and even in the same line. We use · and (·,·) for the usual norm and the inner product ofL(Rn), respectively. We denote by ·
pthe norm ofLp(Rn) (≤p≤ ∞) and by · H
the norm ofH(Rn). In general,m(e) is the Lebesgue measure ofe⊂Rn. ·
Edenotes the
norm of any Banach spaceEandB(E) is the set of all bounded subsets ofE. LetX,Y⊂E, denote by dist (X,Y) =supx∈Xinfy∈Y d(x,y) the semidistance betweenXandY.
2 Preliminaries
In this section, we first recall the basic definitions and theorems.
We say that{U(t,τ)}τ≤tis a continuous process (or norm-to-weak continuous process) in
Xif
() U(t,s)U(s,τ) =U(t,τ),∀t≥s≥τ, () U(τ,τ) =Idis the identity operator,τ∈R, () x→U(t,τ)xis continuous inX
(orU(t,τ)xn U(t,τ)xifxn→x,∀t≥τ,τ∈R).
Suppose thatDis a nonempty class of parameterised setsDˆ ={D(t) :t∈R} ⊂B(E).
Definition .([]) The process{U(t,τ)}τ≤t is said to beD-pullback asymptotically
compact if, for any t∈Rand anyDˆ ∈D, and any sequenceτn→–∞, any sequence
xn∈D(τn), the sequence{U(t,τn)xn}is precompact inX.
Definition . ([]) It is said that Bˆ ∈D is D-pullback absorbing for the process
{U(t,τ)}τ≤t if, for anyt∈Rand anyDˆ ∈D, there existsτ(t,Dˆ)≤tsuch thatU(t,τ)×
D(τ)⊂B(t) for allτ ≤τ(t,Dˆ).
Definition .([]) The familyAˆ ={A(t) :t∈R} ⊂B(E) is said to be aD-pullback at-tractor forU(t,τ) if
() A(t)is compact for allt∈R, () Aˆ is invariant, i.e.
U(t,τ)A(τ) =A(τ) for allt≥τ,
() Aˆ isD-pullback attracting, i.e.
lim τ→–∞dist
U(t,τ)D(τ),A(t)= for allDˆ ∈Dand allt∈R,
() if{C(t)}t∈Ris another family of closed attracting sets, thenA(t)⊂C(t)for allt∈R.
Definition .([]) LetMbe a metric space andAbe a bounded subset ofM. The
Ku-ratowski measure of non-compactnessα(A) is defined by
α(A) =inf{δ> |Aadmits a finite cover by sets of diameter≤δ}.
It has the following properties.
Lemma .([]) Let B,B,B∈B(E).Then () α(B) = ⇔α(N(B,ε))≤ε⇔ ¯Bis compact; () α(B+B)≤α(B) +α(B);
() α(B)≤α(B)wheneverB⊂B; () α(B∪B)≤max{α(B),α(B)}; () α(B) =α(B¯);
() ifBis a ball of radiusε,thenα(B)≤ε.
Definition .([]) A process{U(t,τ)}τ≤tis calledD-pullbackω-limit compact if for
Theorem .([]) Let{U(t,τ)}τ≤t be a process on X.Then{U(t,τ)}τ≤t isD-pullback
asymptotically compact if and only if{U(t,τ)}τ≤tisD-pullbackω-limit compact. Theorem . ([]) Let {U(t,τ)}τ≤t be a norm-to-weak continuous process such that {U(t,τ)}τ≤tisD-pullbackω-limit compact.If there exists a family ofD-pullback
absorb-ing sets{B(t) :t∈R} ∈D,i.e.for any t∈RandDˆ ∈D,there existsτ(t,Dˆ)≤t such that
U(t,τ)D(τ)⊂B(t)for allτ ≤τ,then there exists aD-pullback attractorA={A(t) :t∈R}
and
A(t) =ω(Bˆ,t) =
s≤t
τ≤s
U(t,τ)B(τ).
Remark Obviously, a continuous process and a weak continuous process are both
norm-to-weak continuous processes.
Theorem .([]) Letbe a domain inRn,{U(t,τ)}
τ≤tbe a process on Lp()and Lq()
(p>q≥)and{U(t,τ)}τ≤tsatisfy the following two assumptions: () {U(t,τ)}τ≤tisD-pullbackω-limit compact inLq();
() for anyε> ,Bˆ∈D,there existM(ε,Bˆ)andτ=τ(ε,Bˆ)≤tsuch that
(|U(t,τ)|≥M)
U(t,τ)uτ
p
dx
p
< –pp+ε for anyu
τ ∈B(τ)andτ≥τ.
Then{U(t,τ)}τ≤tisD-pullbackω-limit compact in Lp().
Theorem .([]) Let X,Y be two Banach spaces with the norms·Xand·Y,
respec-tively.Let{U(t,τ)}τ≤t be a continuous process on X and a process on Y.Assume that the
familyBˆ ={B(t) :t∈R}is(X,X)-D-pullback absorbing for U(t,τ),and for any t∈Rand
any sequenceτn→–∞,any sequence xn∈B(τn),the sequence{U(t,τn)xn}is precompact
in X.Then the family of setsA={A(t) :t∈R},where
A(t) =
s≤t
τ≤s
U(t,τ)B(τ)
X
is a(X,X)-D-pullback attractor for{U(t,τ)}τ≤t,where A X
denotes the closure of A with respect to the norm topology in X.
Furthermore, if the family Bˆ ={B(t) :t ∈ R} is (X,Y)-D-pullback absorbing for {U(t,τ)}τ≤t, and it satisfies that, for any t ∈R and any sequence τn→ –∞, any
se-quence xn∈B(τn),the sequence{U(t,τn)xn}is precompact in Y.Then the family of sets
A={A(t) :t∈R},where
A(t) =
s≤t
τ≤s
U(t,τ)B(τ)∩B(τ)
X
=
s≤t
τ≤s
U(t,τ)B(τ)∩B(τ)
Y
is a(X,Y)-D-pullback attractors for{U(t,τ)}τ≤t.
Remark When{U(t,τ)}τ≤tis only a process onY, we also proveA={A(t) :t∈R}is a
Lemma . Let{U(t,τ)}τ≤tbe a process on Lp(Rn) (p≥),Bˆ ={B(t) :t∈R}is(X,Y) -D-pullback absorbing for{U(t,τ)}τ≤t.Then,for anyε> ,t∈RandDˆ ∈D⊂B(Lp(Rn)),
there exist M(t,ε)andτ=τ(t,ε)such that
mRnU(t,τ)≥M(t,ε)<ε for all uτ ∈D(τ)andτ≤τ.
The proof of the above lemma is identical to the proof of Lemma . in [].
Using the standard Faedo-Galerkin method (see [, ]), it is easy to prove the following lemma.
Lemma . Assume that(.)-(.)hold and g∈L
loc(R,L(Rn)).Then,for any T> ,uτ ∈
L(Rn),τ∈Rand T≥τ,there exists a unique weak solution u(x,t)for Eq. (.)satisfying
u∈C[τ,T];LRn∩Lpτ,T;LpRn∩Lτ,T;HRn.
Furthermore,uτ→u(t,τ;uτ)is continuous in L(Rn).
Based on Lemma ., we can define a continuous process{U(t,τ)}τ≤tinL(Rn) by
U(t,τ)uτ=u(t) for allt≥τ, (.)
whereu(t) is the solution of Eq. (.) with the initial valueu(x,τ) =uτ ∈L(Rn). Moreover, we also know that{U(t,τ)}τ≤tis a process inLp(Rn).
3 Main results
3.1 (L2(Rn),L2(Rn))-D-pullback attractors
Firstly, the following lemma ensures aD-pullback absorbing set inL(Rn). Lemma . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))satisfies
(.).Then,for anyDˆ ∈D⊂B(L(Rn))and any t∈R,there existsτ
(t,Dˆ)≤t such that
U(t,τ)uτ≤R(t) for allτ≤τ(t,Dˆ)and all uτ ∈D(τ), (.)
where R(t) = (βσa(x) +
e–σt λ–β
t
–∞eσrg(x,r)dr)
.
Proof Taking the inner product of (.) withuinL(Rn), we have
d dtu
+ν∇u+λu+f (u),u
+a(x)f(u),u
=g(x,t),u. Due to (.)-(.) and Young’s inequality, we get
d dtu
+ (λ–β
)u≤βa(x)+
g(x,t) λ–β
, (.)
d dtu
+ (λ–β
)u+ ν∇u+ αupp+ α
Rna(x)|u|
pdx
≤βa(x)+
g(x,t) λ–β
By (.), we obtain
d dr
eσru+ (λ–β
–σ)eσru≤βa(x)eσr+
g(x,r)
λ–β
eσr.
Integrating over the interval [τ,t] and noting thatσ∈(,λ–β), we have
eσtu(t)≤βa(x)
σ e
σt+
λ–β
t
τ
eσrg(x,r)dr+eσ τuτ
≤βa(x)
σ e
σt+
λ–β
t
–∞e
σrg(x,r)dr+eσ τ
uτ. (.)
Thus, we get
u(t)≤βa(x)
σ +e
–σteσ τu τ+
e–σt
λ–β
t
–∞
eσrg(x,r)
dr,
and this implies (.).
LetBˆ ={B(t) :t∈R}, where
B(t) =
u∈LRn:u ≤R(t)
. (.)
By Lemma ., it is easy to know that the familyBˆ is (L(Rn),L(Rn))-D-pullback ab-sorbing for the process{U(t,τ)}τ≤tdefined by (.) and
eσtR(t)
→
ast→–∞. (.)
LetF(u) =
u
f(s)dsandF(u) =
u
f(s)ds. By (.)-(.), there exist positive constants ˜
αi,β˜i,i= , , , , such that ˜
α|u|p–β˜|u|≤F(u)≤ ˜α|u|p+β˜|u|, λ> β˜, (.) ˜
α|u|p–β˜≤F(u)≤ ˜α|u|p+β˜. (.) Lemma . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))satisfies
(.).Then,for anyDˆ ∈D⊂B(L(Rn))and any t∈R,there existsτ
(t,Dˆ)≤t such that
u(t)+∇u(t)+u(t)pp≤R(t)
for allτ≤τ(t,Dˆ)and all uτ ∈D(τ), (.)
where R(t) =C(βa(x)σ +
e–σt
λ–β
t
–∞eσrg(x,r)dr)
and the positive constant C is
inde-pendent of t andDˆ.
Proof Multiplying (.) byeσt, we have
d dt
eσtu(t)+ (λ–β–σ)eσtu(t)+ νeσt∇u(t)
+ αeσtupp+ αeσt
Rna(x)|u|
pdx
≤βeσta(x)+eσt
g(x,t) λ–β
Letτ<t– andr∈[τ,t– ], integrating over the interval [r,r+ ], we get
eσ(r+)u(r+ )+ (λ–β–σ)
r+
r
eσsu(s)ds+ ν r+
r
eσs∇u(s)ds
+ α
r+
r
eσsu(s)ppds+ α
r+
r
eσs
Rna(x)
u(s)pdx ds
≤βa(x)
r+
r
eσsds+
r+
r
eσsg(x,s)
λ–β
ds+eσru(r)
.
By (.), we find r+
r
eσsu(s)+∇u(s)+u(s)pp+
Rna(x)
u(s)pdx
ds
≤C
βa(x)
σ e
σ(r+)+
λ–β
r+
τ
eσsg(x,s)ds+eσ τuτ
≤C
βa(x)
σ e
σt+eσ τ
uτ+ λ–β
t
–∞e
σsg(x,s)
ds
.
Thus, by (.) and (.), we can obtain
r+
r
eσs
ν ∇u
+λ
u
+
Rn
F(u)dx+
Rn
a(x)F(u)dx
ds
≤C
βa(x)
σ e
σt+eσ τ
uτ+ λ–β
t
–∞
eσsg(x,s)ds
. (.)
Multiplying (.) byut and integrating onRn, we have ut+
d dt
ν ∇u
+λ
u
+
RnF(u)dx+
Rna(x)F(u)dx
≤
g(x,t)
+ut
. And then d dt ν ∇u
+λ
u
+
Rn
F(u)dx+
Rn
a(x)F(u)dx
≤
g(x,t)
. (.)
It follows from (.) that
d dre
σr
ν ∇u
+λ
u
+
RnF(u)dx+
Rna(x)F(u)dx
≤σeσr
ν ∇u
+λ
u
+
RnF(u)dx+
Rna(x)F(u)dx
+e σr
g(x,t)
.
By (.), (.), (.), (.) and the uniform Gronwall inequality, we obtain
ν ∇u
+λ
u
+
RnF(u)dx+
Rna(x)F(u)dx
≤C
βa(x)
σ +e
–σ(t–τ)u
τ+
e–σt
λ–β
t
–∞
eσsg(x,s)
ds
It follows from (.) and (.) that
u(t)+∇u(t)+u(t)pp
≤C
βa(x)
σ +e
–σ(t–τ)u
τ+
e–σt
λ–β
t
–∞e
σsg(x,s)
ds
,
and this implies (.).
Lemma . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))satisfies
(.).Let the familyBˆ ={B(t) :t∈R}be defined by(.).Then,for anyε≥and any
t∈R,there existk˜=k˜(t,ε) > andτ(t,ε)such that
|x|≥k
U(t,τ)uτ
dx≤ε for all k≥ ˜k,τ≤τ(t,ε)and uτ∈B(τ). (.)
Proof Choose a smooth functionθ such that ≤θ(s)≤ fors∈R+,
θ(s) = ⎧ ⎨ ⎩
, ≤s≤, , s≥,
and there exists a constantcsuch that|θ(s)| ≤c. Multiplying (.) byθ(|x|
k )uand integrating onRn, we have
d dt Rn θ |
x| k
|u|dx–ν
Rn θ
|
x| k
uu dx+λ
Rn θ
|
x| k
|u|dx
= –
Rn θ
|
x|
k
f(u)u dx–
Rn θ
|
x|
k
a(x)f(u)u dx+
Rn θ
|
x|
k
g(x,t)u dx
≤β
Rn θ
|
x| k
|u|dx–α
Rn θ
|
x| k
|u|pdx+β
Rn θ
|
x| k
a(x)dx
–α
Rn θ
|x|
k
a(x)|u|pdx+λ–β
Rn θ
|x|
k
|u|dx
+
(λ–β)
Rn θ
|
x| k
g(x,t)dx.
And so d dt Rn θ |
x|
k
|u|dx– ν
Rn θ
|
x|
k
uu dx+ (λ–β)
Rn θ
|
x|
k
|u|dx
≤β
Rn θ
|
x| k
a(x)dx+ (λ–β)
Rn θ
|
x| k
g(x,t)dx. (.)
For the second term on the left-hand side of (.), we know
–ν
Rn θ
|
x| k
uu dx
=ν
Rn θ
|
x|
k
|∇u|dx+ν
Rn θ
|
x|
k
θ
|
x|
k
ux
and Rnθ
|x|
k
θ
|
x|
k
ux k · ∇u dx
≤ √ c k
k≤|x|≤√k|
u||∇u|dx≤C
ku∇u. (.)
It follows from (.) and (.) that
d dr
eσr
Rn θ
|x|
k
|u|dx
≤βeσr
|x|≥k
a(x)dx+ e σr
(λ–β)
|x|≥k
g(x,t)dx+C
ke
σru∇u.
Integrating over the interval [τ,t], we get
Rn θ
|x|
k
u(t)dx
≤βe–σt
t
τ
eσr
|x|≥k
a(x)dx dr+ e
–σt
(λ–β)
t
τ
eσr
|x|≥k
g(x,r)dx dr
+C
ke
–σt
t
τ
eσru∇udr+e–σteσ τu
τ, (.)
whereτ ≤τ(t,Dˆ). We now treat each term on the right-hand side of (.). For the first
term,
βe–σt
t
τ
eσr
|x|≥k
a(x)dx dr≤βe–σteσt
|x|≥k
a(x)dx≤β
|x|≥k
a(x)dx,
by (.), for anyε> , there existsk(ε,t) such that
β
|x|≥k
a(x)dx< ε
for allk≥k. (.)
For the second term, by (.), for anyε> , there existsk(ε,t) such that
(λ–β)
t
τ
eσr
|x|≥k
g(x,r)dx dr
≤
(λ–β)
t
–∞
eσr
|x|≥k
g(x,r)dx dr< ε
for allk≥k. (.)
For the forth term, sinceuτ ∈B(τ), by (.), for anyt∈R, we get
e–σteσ τu
τ→ asτ→–∞. (.)
We now handle the third term on the right-hand side of (.). By Young’s inequality, we know
C ke
–σt
t
τ
eσru∇udr≤ C
ke
–σt
t
τ
eσrudr+ C ke
–σt
t
τ
We can findδ> such that
t
τ
eσrudr≤
t
τ
e(σ+δ)rudr,
and by (.), we have t
τ
e(σ+δ)ru(r)dr
≤
t
τ
e(σ+δ)r
βa(x)
σ +e
–σreσ τu τ+
e–σr
λ–β
r
–∞e
σsg(x,s)
ds
dr
≤βa(x)
σ e
(σ+δ)t+eσ τ
uτ t
τ
eδrdr+
λ–β
t
τ
eδsds
t
–∞
eσsg(x,s)ds
≤βa(x)
σ e
(σ+δ)t+
δ
eδteσ τu τ+
δ(λ–β)
eδt
t
–∞e
σsg(x,s)
ds
<∞.
Analogously, we can obtain t
τ
eσr∇udr<∞.
Thus, for anyε> , there existsk(ε,t) such that
C ke
–σt
t
τ
eσru∇udr<ε
for allk≥k. (.)
It follows from (.)-(.) that
|x|≥k
U(t,τ)uτ
dx≤
Rn θ
|
x|
k
u(t)dx<ε.
So, the proof is complete.
Next, we utilise Definition . to prove that the process{U(t,τ)}τ≤tassociated with the
initial value problem (.) isD-pullbackω-limit compact.
Lemma . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))satisfies
(.).Then the process{U(t,τ)}τ≤t associated with the initial value problem (.)isD
-pullbackω-limit compact in L(Rn).
Proof DenoteBr=B(,r)∩Rn, we can splitu(t) as
u(t) =χ(x)u(t) + –χ(x)u(t),
whereχ(x) is a smooth function satisfying ≤χ(x)≤,|χ(x)| ≤c, and it is defined by
χ(x) = ⎧ ⎨ ⎩
, x∈Br,
And so, we have
u(t) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
u(t), x∈Br,
, x∈/Br+,
χ(x)u(t), others,
u(t) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
, x∈Br,
u(t), x∈/Br+,
( –χ(x))u(t), others. For anyDˆ ∈D⊂B(L(Rn)),{U(t,τ)D(τ)}={U(t,τ)u
τ |uτ ∈D(τ)}can be split as
U(t,τ)D(τ) =χ(x)U(t,τ)D(τ) + –χ(x)U(t,τ)D(τ). By Lemma ., we have
αU(t,τ)D(τ)≤αχ(x)U(t,τ)D(τ)+α –χ(x)U(t,τ)D(τ). (.) By Lemma ., we getu(t)∈L(Br) asτ≤τ(t,Dˆ) and
χ(x)U(t,τ)D(τ) =χ(x)U(t,τ)uτ =u(t)|uτ ∈D(τ)
.
By Lemma ., we have u(t)
H(Br+)=∇u(t)L(Br+)≤R(t) for allτ≤τ(t,Dˆ).
SinceH
(Br+)→L(Br+) is compact,χ(x)U(t,τ)D(τ) is compact inL(Br+). By
Lem-ma ., we obtain
αχ(x)U(t,τ)D(τ)= . (.)
By Lemma ., for anyε> , we can chooserlarge enough such that
|x|≥r
|u|≤ε.
And then
u ≤ε for allτ≤τ(t,ε). (.)
We know
–χ(x)U(t,τ)D(τ) = –χ(x)U(t,τ)uτ =u(t)|uτ ∈D(τ)
.
By (.), we obtain
α –χ(x)U(t,τ)D(τ)≤ε for allτ ≤τ(t,ε). (.) It follows from (.), (.) and (.) that
αU(t,τ)D(τ)≤ε for allτ≤minτ(t,Dˆ),τ(t,Dˆ),τ(t,ε)
.
Using Theorem . or Theorem ., it is easy to prove the following theorem by Lemma . and Lemma ..
Theorem . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))
satis-fies(.).Then the process{U(t,τ)}τ≤tassociated with the initial value problem(.)has a D-pullback attractorA={A(t) :t∈R}in L(Rn).
3.2 (L2(Rn),Lp(Rn))-D-pullback attractors
In this subsection, we prove the existence ofD-pullback attractors inLp(Rn). We setBˆ = {B(t) :t∈R}, where
B(t) =
u∈LRn∩LpRn:u+up≤R(t)
for allt∈R, (.) andR(t) is defined in Lemma .. So by Lemma ., we obtain the familyBˆ ={B(t) :t∈
R}is (L(Rn),Lp(Rn))-D-pullback absorbing for the process{U(t,τ)}
τ≤t, i.e. for anyDˆ ∈ D⊂B(L(Rn)), there existsτ(t,Dˆ)≤tsuch thatU(t,τ)D(τ)⊂B(t) for allτ≤τ(t,Dˆ).
We also know
eσtR (t)
→ ast→–∞. (.)
Based on Theorem ., we only prove that the process {U(t,τ)}τ≤t associated with the
initial value problem (.) isD-pullbackω-limit compact inLp(Rn). Firstly, we prove the
following lemma.
Lemma . Assume that(.)-(.)hold and the external force g∈L
loc(R,L(Rn))satisfies
(.).Let the familyBˆ={B(t) :t∈R}be defined by(.).Then,for anyε≥,anyDˆ ∈ D⊂B(L(Rn))and any t∈R,there exist M=M(t,ε) > andτ(t,ε)such that
Rn(|U(t,τ)u τ|≥M)
U(t,τ)uτ
p
dx≤ε
for all uτ ∈D(τ),τ≤τ(t,ε)and M≥M. (.)
Proof For anyε> be given, by (.), there existsδ> such that
t
–∞
eσs
e
g(x,s)dx ds<ε, (.)
wheree⊂Rnandm(e)≤δ. By Lemma . and Lemma ., we know that there exist
M=M(t,ε) andτ=τ(t,ε) such that
mRnU(t,τ)uτ≥M
≤δ for alluτ ∈D(τ) andτ≤τ. (.)
By (.) and (.), we can chooseMlarge enough such that
α|u|p––β|u| ≤f(u)≤α|u|p–+β|u| inRn
U(t,τ)uτ ≥M
, (.)
α|u|p–≤f(u)≤α|u|p– inRn
U(t,τ)uτ≥M
LetM=max{M,M}andτ≤τ. Multiplying Eq. (.) by (u–M)p–+ and integrating on
Rn, we have
p d dt
Rn
(u–M)+ p
dx–ν
Rn
u(u–M)p–+ dx+λ
Rn
u(u–M)p–+ dx
+
Rnf
(u)(u–M)p–+ dx+
Rna
(x)f(u)(u–M)p–+ dx=
Rng
(x,t)(u–M)p–+ dx,
where (u–M)+denotes the positive part ofu–M, that is
(u–M)+=
⎧ ⎨ ⎩
u–M, u≥M,
, u<M.
Let=Rn(U(t,τ)uτ ≥M), we get
(u–M)p–+ ≤ |u|p–(u–M)p–+ and (u–M)p+≤u(u–M)p–+ in.
It follows from (.), (.), Young’s inequality and Hölder’s inequality that
–ν
Rn
u(u–M)p–+ dx=ν(p– )
|∇u|(u–M)p–+ dx≥, (.)
Rnf
(u)(u–M)p–+ dx≥
α|u|p–(u–M)p–+ dx
–
β|u|(u–M)p–+ dx, (.)
Rn
a(x)f(u)(u–M)p–+ dx≥α
a(x)|u|p–(u–M)p–+ dx≥, (.)
Rng(x,t)(u–M)
p–
+ dx≤
α
g(x,t)dx+α
(u–M)p–+ dx ≤
α
g(x,t)dx+α
|u|p–(u–M)p–+ dx. (.)
By (.)-(.), we get
p d dt
(u–M)+pdx+ (λ–β)
(u–M)+pdx≤
α
g(x,t)dx,
which implies that
d dt(t–τ)e
σt
(u–M)+pdx+ceσt
(u–M)+pdx
≤p(t–τ)
α
eσt
g(x,t)dx, (.)
whereu> inandc= (p(λ–β) –σ)(t–τ) – . Sinceσ∈(,λ–β) andp> , there
existsτ=τ(t,ε) < such that
p(λ–β) –σ
So integrating (.) over the interval [τ,t], we have
(u–M)+pdx≤
p
α
e–σt
t
–∞e σs
g(x,s)dx ds.
By (.), we can obtain
(u–M)+pdx≤Cε for allτ ≤τanduτ∈D(τ), (.)
whereC> is a constant independent ofM. Set=Rn(U(t,τ)uτ ≤–M). Likewise,
replacing (u–M)+with (u+M)–, we can also obtain that there existsτ=τ(t,ε) such
that
(u+M)– p
dx≤Cε for allτ≤τanduτ∈D(τ), (.)
where (u+M)–is the negative part ofu+M, that is
(u+M)–=
⎧ ⎨ ⎩
u+M, u≤–M,
, u> –M.
Then it follows from (.) and (.) that
Rn(|U(t,τ)uτ|≥M)
|u|–M p
dx≤ε for allτ≤τ(t,ε) anduτ∈D(τ),
whereτ(t,ε) =min{τ,τ}. Hence, we get
Rn(|U(t,τ)u τ|≥M)
U(t,τ)uτpdx
=
Rn(|U(t,τ)uτ|≥M)
|u|–M+M
p
dx
≤p–
Rn(|U(t,τ)u τ|≥M)
|u|–M
p
dx+
Rn(|U(t,τ)u τ|≥M)
(M)pdx
≤p–
Rn(|U(t,τ)u τ|≥M)
|u|–M
p
dx+
Rn(|U(t,τ)u τ|≥M)
|u|–M
p
dx
≤pε.
Finally, we obtain (.) and the proof is complete.
By Theorem ., Lemma . and Lemma ., we can obtain that the process{U(t,τ)}τ≤t
associated with the initial value problem (.) isD-pullbackω-limit compact inLp(Rn).
So it is easy to prove the following theorem.
Theorem . Assume that(.)-(.)hold and the external force g∈Lloc(R,L(Rn))
satis-fies(.).Then the family of setsA={A(t) :t∈R}is(L(Rn),Lp(Rn))-D-pullback
Proof We know that the familyBˆ ={B(t) :t∈R}is (L(Rn),Lp(Rn))-D-pullback
absorb-ing for the process{U(t,τ)}τ≤t, whereB(t) is defined by (.). Thus, by Theorem ., we
can deduce that the theorem is true.
Funding Not applicable.
Abbreviations Not applicable.
Availability of data and materials Not applicable.
Ethics approval and consent to participate Not applicable.
Competing interests
The author declares that they have no competing interests.
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Authors’ contributions
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Received: 7 August 2017 Accepted: 25 September 2017
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