Zhang Boundary Value Problems (2017) 2017:146 DOI 10.1186/s13661-017-0879-5
RESEARCH
Open Access
Pullback attractors for a class of non-autonomous reaction-diffusion equations in Rn 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 in Rn , where the external force g(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 nonautonomous reaction-diffusion equations defined in the whole space: ⎧ ⎨u – νu + λu + f (u) + a(x)f (u) = g(x, t), in Rn × [τ , ∞), t (.) ⎩u(x, τ ) = uτ , in Rn , where ν and λ are positive constants. Assume that nonlinear terms f (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
(.)
with p > and λ > β , α |u|p – β ≤ f (u)u ≤ α |u|p + β
and
f (u) ≥ –l
(.)
with p > , where αi , βi , i = , , , , and li , i = , are positive constants. Furthermore, a(x) is a function in Rn and the external force g(x, t) ∈ Lloc (R; L (Rn )) satisfies the following conditions: a(x) ∈ L Rn ∩ L∞ Rn and a(x) > , t eσ s g(x, s)L (Rn ) ds < ∞, for all t ∈ R, σ ∈ (, λ – β ).
(.) (.)
–∞
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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In the last decade, the autonomous and non-autonomous infinite dimensional dynamical systems have been studied extensively by many authors (see, e.g. [–] and the references 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∈R of 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 nonautonomous dynamical systems and random dynamical systems (see, e.g. [–] and the references therein). In [], the authors introduced the notion of D-pullback attractors, which requires that the process U(t, τ ) associated with the systems be D-pullback asymptotically compact. It is well known that the Sobolev embeddings are no longer compact in unbounded domain, and so it is difficult to verify the process U(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 in L (Rn ) and H (Rn ) for non-autonomous reaction-diffusion equations defined on Rn . Recently, motivated by [], the authors of [] gave a new method to prove the existence of D-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 pullback 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 nonlinearity f (u) as given in [] that f (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 of D-pullback attractors in L (Rn ) for Eq. (.) and, motivated by the idea in [, ], we obtain the existence of D-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 letter C denotes 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 of L (Rn ), respectively. We denote by · p the norm of Lp (Rn ) ( ≤ p ≤ ∞) and by · H the norm of H (Rn ). In general, m(e) is the Lebesgue measure of e ⊂ Rn . · E denotes the norm of any Banach space E and B(E) is the set of all bounded subsets of E. Let X, Y ⊂ E, denote by dist (X, Y ) = supx∈X infy∈Y d(x, y) the semidistance between X and Y .
2 Preliminaries In this section, we first recall the basic definitions and theorems. Definition . ([]) Let X be a complete metric space and {U(t, τ )} = {U(t, τ ) : t ≥ τ , τ ∈ R} be a two-parameter family of mappings acting on X: U(t, τ ) : X → X, t ≥ τ , τ ∈ R.
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We say that {U(t, τ )}τ ≤t is a continuous process (or norm-to-weak continuous process) in X if () U(t, s)U(s, τ ) = U(t, τ ), ∀t ≥ s ≥ τ , () U(τ , τ ) = Id is the identity operator, τ ∈ R, () x → U(t, τ )x is continuous in X (or U(t, τ )xn U(t, τ )x if xn → x, ∀t ≥ τ , τ ∈ R). Suppose that D is a nonempty class of parameterised sets Dˆ = {D(t) : t ∈ R} ⊂ B(E). Definition . ([]) The process {U(t, τ )}τ ≤t is said to be D-pullback asymptotically compact if, for any t ∈ R and any Dˆ ∈ D, and any sequence τn → –∞, any sequence xn ∈ D(τn ), the sequence {U(t, τn )xn } is precompact in X. Definition . ([]) It is said that Bˆ ∈ D is D-pullback absorbing for the process ˆ ≤ t such that U(t, τ ) × {U(t, τ )}τ ≤t if, for any t ∈ R and any Dˆ ∈ D, there exists τ (t, D) ˆ D(τ ) ⊂ B(t) for all τ ≤ τ (t, D). ˆ = {A(t) : t ∈ R} ⊂ B(E) is said to be a D-pullback atDefinition . ([]) The family A tractor for U(t, τ ) if () A(t) is compact for all t ∈ R, ˆ is invariant, i.e. () A U(t, τ )A(τ ) = A(τ ) for all t ≥ τ , ˆ is D-pullback attracting, i.e. () A lim dist U(t, τ )D(τ ), A(t) =
τ →–∞
for all Dˆ ∈ D and all t ∈ R,
() if {C(t)}t∈R is another family of closed attracting sets, then A(t) ⊂ C(t) for all t ∈ R. Definition . ([]) Let M be a metric space and A be a bounded subset of M. The Kuratowski measure of non-compactness α(A) is defined by α(A) = inf{δ > | A admits 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 ⇔ B¯ is compact; () α(B + B ) ≤ α(B ) + α(B ); () α(B ) ≤ α(B ) whenever B ⊂ B ; () α(B ∪ B ) ≤ max{α(B ), α(B )}; ¯ () α(B) = α(B); () if B is a ball of radius ε, then α(B) ≤ ε. Definition . ([]) A process {U(t, τ )}τ ≤t is called D-pullback ω-limit compact if for ˆ ≤ t such that α( τ ≤τ U(t, τ )D(τ )) ≤ ε. any ε > and Dˆ ∈ D, there exists τ (t, D)
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Theorem . ([]) Let {U(t, τ )}τ ≤t be a process on X. Then {U(t, τ )}τ ≤t is D-pullback asymptotically compact if and only if {U(t, τ )}τ ≤t is D-pullback ω-limit compact. Theorem . ([]) Let {U(t, τ )}τ ≤t be a norm-to-weak continuous process such that {U(t, τ )}τ ≤t is D-pullback ω-limit compact. If there exists a family of D-pullback absorbˆ ≤ t such that ing sets {B(t) : t ∈ R} ∈ D, i.e. for any t ∈ R and Dˆ ∈ D, there exists τ (t, D) U(t, τ )D(τ ) ⊂ B(t) for all τ ≤ τ , then there exists a D-pullback attractor A = {A(t) : t ∈ R} and ˆ t) = A(t) = ω(B,
U(t, τ )B(τ ).
s≤t τ ≤s
Remark Obviously, a continuous process and a weak continuous process are both normto-weak continuous processes. Theorem . ([]) Let be a domain in Rn , {U(t, τ )}τ ≤t be a process on Lp ( ) and Lq ( ) (p > q ≥ ) and {U(t, τ )}τ ≤t satisfy the following two assumptions: () {U(t, τ )}τ ≤t is D-pullback ω-limit compact in Lq ( ); ˆ ≤ t such that ˆ and τ = τ (ε, B) () for any ε > , Bˆ ∈ D, there exist M(ε, B)
U(t, τ )uτ p dx
p
< –
p+ p
ε
for any uτ ∈ B(τ ) and τ ≥ τ .
(|U(t,τ )|≥M)
Then {U(t, τ )}τ ≤t is D-pullback ω-limit compact in Lp ( ). Theorem . ([]) Let X, Y be two Banach spaces with the norms · X and · Y , respectively. Let {U(t, τ )}τ ≤t be a continuous process on X and a process on Y . Assume that the family Bˆ = {B (t) : t ∈ R} is (X, X)-D-pullback absorbing for U(t, τ ), and for any t ∈ R and any sequence τn → –∞, any sequence xn ∈ B (τn ), the sequence {U(t, τn )xn } is precompact in X. Then the family of sets A = {A(t) : t ∈ R}, where A(t) =
X
U(t, τ )B(τ )
s≤t τ ≤s X
is a (X, X)-D-pullback attractor for {U(t, τ )}τ ≤t , where A 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 sequence 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
X Y U(t, τ ) B (τ ) ∩ B (τ ) = U(t, τ ) B (τ ) ∩ B (τ ) s≤t τ ≤s
is a (X, Y )-D-pullback attractors for {U(t, τ )}τ ≤t . Remark When {U(t, τ )}τ ≤t is only a process on Y , we also prove A = {A (t) : t ∈ R} is a (X, Y )-D-pullback attractor for {U(t, τ )}τ ≤t .
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Lemma . Let {U(t, τ )}τ ≤t be 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 any ε > , t ∈ R and Dˆ ∈ D ⊂ B(Lp (Rn )), there exist M(t, ε) and τ = τ (t, ε) such that
m Rn U(t, τ ) ≥ M(t, ε) < ε
for all uτ ∈ D(τ ) and τ ≤ 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 ∈ Lloc (R, L (Rn )). Then, for any T > , uτ ∈ L (Rn ), τ ∈ R and T ≥ τ , there exists a unique weak solution u(x, t) for Eq. (.) satisfying u ∈ C [τ , T]; L Rn ∩ Lp τ , T; Lp Rn ∩ L τ , T; H Rn . Furthermore, uτ → u(t, τ ; uτ ) is continuous in L (Rn ). Based on Lemma ., we can define a continuous process {U(t, τ )}τ ≤t in L (Rn ) by U(t, τ )uτ = u(t) for all t ≥ τ ,
(.)
where u(t) is the solution of Eq. (.) with the initial value u(x, τ ) = uτ ∈ L (Rn ). Moreover, we also know that {U(t, τ )}τ ≤t is a process in Lp (Rn ).
3 Main results 3.1 (L2 (Rn ), L2 (Rn ))-D-pullback attractors Firstly, the following lemma ensures a D-pullback absorbing set in L (Rn ). Lemma . Assume that (.)-(.) hold and the external force g ∈ Lloc (R, L (Rn )) satisfies ˆ ≤ t such that (.). Then, for any Dˆ ∈ D ⊂ B(L (Rn )) and any t ∈ R, there exists τ (t, D) U(t, τ )uτ ≤ R (t) for all τ ≤ τ (t, D) ˆ and all uτ ∈ D(τ ), where R (t) = ( β a(x) + σ
e–σ t λ–β
(.)
t
σr –∞ e g(x, r) dr) .
Proof Taking the inner product of (.) with u in L (Rn ), we have d u + ν∇u + λu + f (u), u + a(x)f (u), u = g(x, t), u . dt Due to (.)-(.) and Young’s inequality, we get g(x, t) d u + (λ – β )u ≤ β a(x) + , dt λ – β d u + (λ – β )u + ν∇u + α upp + α dt g(x, t) . ≤ β a(x) + λ – β
(.) Rn
a(x)|u|p dx (.)
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By (.), we obtain g(x, r) σ r d σr e u + (λ – β – σ )eσ r u ≤ β a(x) eσ r + e . dr λ – β Integrating over the interval [τ , t] and noting that σ ∈ (, λ – β ), we have β a(x) σ t e + eσ t u(t) ≤ σ λ – β ≤
β a(x) σ t e + σ λ – β
t
eσ r g(x, r) dr + eσ τ uτ
τ
t
eσ r g(x, r) dr + eσ τ uτ .
(.)
–∞
Thus, we get –σ t u(t) ≤ β a(x) + e–σ t eσ τ uτ + e σ λ – β
t
eσ r g(x, r) dr,
–∞
and this implies (.). Let Bˆ = {B (t) : t ∈ R}, where B (t) = u ∈ L Rn : u ≤ R (t) .
(.)
By Lemma ., it is easy to know that the family Bˆ is (L (Rn ), L (Rn ))-D-pullback absorbing for the process {U(t, τ )}τ ≤t defined by (.) and eσ t R (t) →
as t → –∞.
(.)
u u Let F (u) = f (s) ds and F (u) = f (s) ds. By (.)-(.), there exist positive constants α˜i , 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 ∈ Lloc (R, L (Rn )) satisfies ˆ ≤ t such that (.). Then, for any Dˆ ∈ D ⊂ B(L (Rn )) and any t ∈ R, there exists τ (t, D) u(t) + ∇u(t) + u(t)p ≤ R (t) p where R (t) = C( β a(x) + σ ˆ pendent of t and D.
e–σ t λ–β
t
ˆ and all uτ ∈ D(τ ), (.) for all τ ≤ τ (t, D)
σr –∞ e g(x, r) dr)
and the positive constant C is inde-
Proof Multiplying (.) by eσ t , we have d σt e u(t) + (λ – β – σ )eσ t u(t) + νeσ t ∇u(t) dt + α eσ t upp + α eσ t a(x)|u|p dx Rn
g(x, t) . ≤ β eσ t a(x) + eσ t λ – β
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Let τ < t – and r ∈ [τ , t – ], integrating over the interval [r, r + ], we get u(r + ) + (λ – β – σ )
σ (r+)
e
r
r+
+ α r
≤ β a(x)
p eσ s u(s)p ds + α
r+
σ s
eσ s
Rn
r
eσ s r
r+
eσ s ∇u(s) ds
r
r+
r+
eσ s ds + r
e u(s) ds + ν
r+
p a(x) u(s) dx ds
g(x, s) ds + eσ r u(r) . λ – β
By (.), we find
r+
r
p eσ s u(s) + ∇u(s) + u(s)p +
≤C
β a(x) σ (r+) e + σ λ – β
r+
Rn
p a(x) u(s) dx ds
eσ s g(x, s) ds + eσ τ uτ
τ
β a(x) σ t σ τ e + e uτ + ≤C σ λ – β
t
e g(x, s) ds . σ s
–∞
Thus, by (.) and (.), we can obtain
r+
eσ s r
λ ν ∇u + u +
Rn
F (u) dx +
β a(x) σ t σ τ ≤C e + e uτ + σ λ – β
Rn t
a(x)F (u) dx ds e g(x, s) ds . σ s
(.)
–∞
Multiplying (.) by ut and integrating on Rn , we have d ν λ ∇u + u + F (u) dx + a(x)F (u) dx dt Rn Rn ≤ g(x, t) + ut .
ut +
And then λ d ν ∇u + u + F (u) dx + a(x)F (u) dx ≤ g(x, t) . dt Rn Rn
(.)
It follows from (.) that λ d σr ν e ∇u + u + F (u) dx + a(x)F (u) dx dr Rn Rn λ eσ r σr ν g(x, t) . ∇u + u + F (u) dx + a(x)F (u) dx + ≤ σe n n R R By (.), (.), (.), (.) and the uniform Gronwall inequality, we obtain λ ν ∇u + u + F (u) dx + a(x)F (u) dx Rn Rn t e–σ t β a(x) + e–σ (t–τ ) uτ + eσ s g(x, s) ds . ≤C σ λ – β –∞
(.)
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It follows from (.) and (.) that u(t) + ∇u(t) + u(t)p p t e–σ t β a(x) + e–σ (t–τ ) uτ + ≤C eσ s g(x, s) ds , σ λ – β –∞
and this implies (.).
Lemma . Assume that (.)-(.) hold and the external force g ∈ Lloc (R, L (Rn )) satisfies (.). Let the family Bˆ = {B (t) : t ∈ R} be defined by (.). Then, for any ε ≥ and any ˜ ε) > and τ (t, ε) such that t ∈ R, there exist k˜ = k(t, |x|≥k
U(t, τ )uτ dx ≤ ε
˜ τ ≤ τ (t, ε) and uτ ∈ B (τ ). for all k ≥ k,
(.)
Proof Choose a smooth function θ such that ≤ θ (s) ≤ for s ∈ R+ , ⎧ ⎨, ≤ s ≤ , θ (s) = ⎩, s ≥ , and there exists a constant c such that |θ (s)| ≤ c. )u and integrating on Rn , we have Multiplying (.) by θ ( |x| k
|x| |x| |x| θ θ θ |u| dx – ν uu dx + λ |u| dx k k k Rn Rn Rn |x| |x| |x| θ θ θ f (u)u dx – a(x)f (u)u dx + g(x, t)u dx =– k k k Rn Rn Rn |x| |x| p |x| θ dx – α θ dx + β θ |u| |u| a(x) dx ≤ β k k k Rn Rn Rn λ – β |x| p |x| θ dx + θ a(x)|u| |u| dx – α k k Rn Rn
|x|
g(x, t) dx. θ + (λ – β ) Rn k
d dt
And so d dt
|x| |x| |x| |u| uu dx + (λ – β |u| dx dx – ν θ ) θ k k k Rn Rn Rn
|x| |x|
g(x, t) dx. θ θ (.) a(x) dx + ≤ β k (λ – β ) Rn k Rn
θ
For the second term on the left-hand side of (.), we know
|x| θ uu dx –ν k Rn |x| |x| x |x| |∇u| θ u · ∇u dx θ dx + ν θ =ν k k k k Rn Rn
(.)
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and
|x| |x| x θ θ u · ∇u dx
k k k Rn √ C c u∇u. ≤ √ |u||∇u| dx ≤ k k k≤|x|≤ k
(.)
It follows from (.) and (.) that d σr |x| |u| e θ dx dr k Rn
eσ r σr
g(x, t) dx + C eσ r u∇u. a(x) dx + ≤ β e (λ – β ) k |x|≥k |x|≥k Integrating over the interval [τ , t], we get
|x|
u(t) dx k Rn t eσ r a(x) dx dr + ≤ β e–σ t
θ
|x|≥k
τ
+
C –σ t e k
t
e–σ t (λ – β )
t
eσ r τ
|x|≥k
g(x, r) dx dr
eσ r u∇u dr + e–σ t eσ τ uτ ,
(.)
τ
ˆ We now treat each term on the right-hand side of (.). For the first where τ ≤ τ (t, D). term,
t
β e–σ t
eσ r
|x|≥k
τ
a(x) dx dr ≤ β e–σ t eσ t
|x|≥k
a(x) dx ≤ β
|x|≥k
a(x) dx,
by (.), for any ε > , there exists k (ε, t) such that β
|x|≥k
a(x) dx <
ε
for all k ≥ k .
(.)
For the second term, by (.), for any ε > , there exists k (ε, t) such that
(λ – β ) ≤
t σr
e τ
(λ – β )
|x|≥k t
eσ r –∞
g(x, r) dx dr
g(x, r) dx dr < ε |x|≥k
for all k ≥ k .
(.)
For the forth term, since uτ ∈ B (τ ), by (.), for any t ∈ R, we get e–σ t eσ τ uτ → as τ → –∞.
(.)
We now handle the third term on the right-hand side of (.). By Young’s inequality, we know C –σ t e k
τ
t
eσ r u∇u dr ≤
C –σ t e k
t τ
eσ r u dr +
C –σ t e k
τ
t
eσ r ∇u dr.
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We can find δ > such that
t
t
eσ r u dr ≤
τ
e(σ +δ )r u dr,
τ
and by (.), we have τ
t
e(σ +δ )r u(r) dr
r e–σ r β a(x) + e–σ r eσ τ uτ + eσ s g(x, s) ds dr σ λ – β –∞ τ t t t β a(x) (σ +δ )t σ τ δ r δ s e + e uτ e dr + e ds eσ s g(x, s) ds ≤ σ λ – β τ τ –∞ t β a(x) (σ +δ )t δ t σ τ e eδ t + e e uτ + eσ s g(x, s) ds ≤ σ δ δ (λ – β ) –∞
≤
t
e(σ +δ )r
< ∞. Analogously, we can obtain
t
eσ r ∇u dr < ∞.
τ
Thus, for any ε > , there exists k (ε, t) such that C –σ t e k
t
eσ r u∇u dr <
τ
ε
for all k ≥ k .
(.)
It follows from (.)-(.) that |x|≥k
U(t, τ )uτ dx ≤
θ
Rn
|x|
u(t) dx < ε. k
So, the proof is complete.
Next, we utilise Definition . to prove that the process {U(t, τ )}τ ≤t associated with the initial value problem (.) is D-pullback ω-limit compact. Lemma . Assume that (.)-(.) hold and the external force g ∈ Lloc (R, L (Rn )) satisfies (.). Then the process {U(t, τ )}τ ≤t associated with the initial value problem (.) is Dpullback ω-limit compact in L (Rn ). Proof Denote Br = B(, r) ∩ Rn , we can split u(t) as u(t) = χ(x)u(t) + – χ(x) u(t), where χ(x) is a smooth function satisfying ≤ χ(x) ≤ , |χ (x)| ≤ c , and it is defined by ⎧ ⎨, x ∈ B , r χ(x) = ⎩, x ∈/ Br+ .
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And so, we have ⎧ ⎪ ⎪ ⎨u(t),
⎧ ⎪ ⎪ ⎨,
x ∈ Br ,
u (t) = , x ∈/ Br+ , ⎪ ⎪ ⎩ χ(x)u(t), others,
x ∈ Br ,
u (t) = u(t), x ∈/ Br+ , ⎪ ⎪ ⎩ ( – χ(x))u(t), others.
For any Dˆ ∈ D ⊂ B(L (Rn )), {U(t, τ )D(τ )} = {U(t, τ )uτ | 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 α U(t, τ )D(τ ) ≤ α χ(x)U(t, τ )D(τ ) + α – χ(x) U(t, τ )D(τ ) .
(.)
ˆ and By Lemma ., we get u (t) ∈ L (Br ) as τ ≤ τ (t, D) 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 (B
r+ )
= ∇u (t)L (B
r+ )
ˆ ≤ R (t) for all τ ≤ τ (t, D).
Since H (Br+ ) → L (Br+ ) is compact, χ(x)U(t, τ )D(τ ) is compact in L (Br+ ). By Lemma ., we obtain α χ(x)U(t, τ )D(τ ) = .
(.)
By Lemma ., for any ε > , we can choose r large enough such that |x|≥r
|u| ≤ ε.
And then u ≤ ε
for all τ ≤ 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 all τ ≤ τ (t, ε).
(.)
It follows from (.), (.) and (.) that α U(t, τ )D(τ ) ≤ ε
ˆ τ (t, D), ˆ τ (t, ε) . for all τ ≤ min τ (t, D),
By Definition ., we obtain {U(t, τ )}τ ≤t is D-pullback ω-limit compact in L (Rn ).
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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 ∈ Lloc (R, L (Rn )) satisfies (.). Then the process {U(t, τ )}τ ≤t associated with the initial value problem (.) has a D-pullback attractor A = {A(t) : t ∈ R} in L (Rn ).
3.2 (L2 (Rn ), Lp (Rn ))-D-pullback attractors In this subsection, we prove the existence of D-pullback attractors in Lp (Rn ). We set Bˆ = {B (t) : t ∈ R}, where B (t) = u ∈ L Rn ∩ Lp Rn : u + up ≤ R (t) for all t ∈ R,
(.)
and R (t) is defined in Lemma .. So by Lemma ., we obtain the family Bˆ = {B (t) : t ∈ R} is (L (Rn ), Lp (Rn ))-D-pullback absorbing for the process {U(t, τ )}τ ≤t , i.e. for any Dˆ ∈ ˆ ≤ t such that U(t, τ )D(τ ) ⊂ B (t) for all τ ≤ τ (t, D). ˆ D ⊂ B(L (Rn )), there exists τ (t, D) We also know eσ t R (t) → as t → –∞.
(.)
Based on Theorem ., we only prove that the process {U(t, τ )}τ ≤t associated with the initial value problem (.) is D-pullback ω-limit compact in Lp (Rn ). Firstly, we prove the following lemma. Lemma . Assume that (.)-(.) hold and the external force g ∈ Lloc (R, L (Rn )) satisfies (.). Let the family Bˆ = {B (t) : t ∈ R} be defined by (.). Then, for any ε ≥ , any Dˆ ∈ D ⊂ B(L (Rn )) and any t ∈ R, there exist M = M(t, ε) > and τ (t, ε) such that Rn (|U(t,τ )uτ |≥M)
U(t, τ )uτ p dx ≤ ε
for all uτ ∈ D(τ ), τ ≤ τ (t, ε) and M ≥ M .
(.)
Proof For any ε > be given, by (.), there exists δ > such that
t
eσ s –∞
g(x, s) dx ds < ε,
(.)
e
where e ⊂ Rn and m(e ) ≤ δ . By Lemma . and Lemma ., we know that there exist M = M (t, ε) and τ = τ (t, ε) such that
m Rn U(t, τ )uτ ≥ M ≤ δ
for all uτ ∈ D(τ ) and τ ≤ τ .
(.)
By (.) and (.), we can choose M large enough such that α |u|p– – β |u| ≤ f (u) ≤ α |u|p– + β |u| in Rn U(t, τ )uτ ≥ M , α |u|p– ≤ f (u) ≤ α |u|p– in Rn U(t, τ )uτ ≥ M .
(.) (.)
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p–
Let M = max{M , M } and τ ≤ τ . Multiplying Eq. (.) by (u – M )+ and integrating on Rn , we have
d p dt
Rn
+
Rn
(u – M )+ p dx – ν
Rn
u(u – M )p– + dx + λ
f (u)(u – M )p– + dx +
Rn
u(u – M )p– + dx
a(x)f (u)(u – M )p– + dx =
Rn
Rn
g(x, t)(u – M )p– + dx,
where (u – M )+ denotes the positive part of u – 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
Rn
u(u – M )p– + dx = ν(p – )
f (u)(u – M )p– + dx ≥
(.)
β |u|(u – M )p– + dx,
(.)
a(x)f (u)(u – M )p– + dx ≥ α
Rn
|∇u| (u – M )p– + dx ≥ ,
α |u|p– (u – M )p– + dx
–
Rn
a(x)|u|p– (u – M )p– + dx ≥ ,
g(x, t) dx + α (u – M )p– dx +
g(x, t) dx + α |u|p– (u – M )p– ≤ + dx. α
g(x, t)(u – M )p– + dx ≤
α
(.)
(.)
By (.)-(.), we get d p dt
(u – M )+ p dx + (λ – β )
(u – M )+ p dx ≤ α
g(x, t) dx,
which implies that d (t – τ )eσ t dt ≤
p(t – τ ) σ t e α
(u – M )+ p dx + c eσ t
(u – M )+ p dx
g(x, t) dx,
(.)
where u > in and c = (p(λ – β ) – σ )(t – τ ) – . Since σ ∈ (, λ – β ) and p > , there exists τ = τ (t, ε) < such that
p(λ – β ) – σ (t – τ ) ≥
for all τ ≤ τ .
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So integrating (.) over the interval [τ , t], we have
(u – M )+ p dx ≤ p e–σ t α
t
g(x, s) dx ds.
eσ s –∞
By (.), we can obtain
(u – M )+ p dx ≤ Cε
for all τ ≤ τ and uτ ∈ D(τ ),
(.)
where C > is a constant independent of M . Set = Rn (U(t, τ )uτ ≤ –M ). Likewise, replacing (u – M )+ with (u + M )– , we can also obtain that there exists τ = τ (t, ε) such that
(u + M )– p dx ≤ Cε
for all τ ≤ τ and uτ ∈ D(τ ),
(.)
where (u + M )– is the negative part of u + 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 all τ ≤ τ (t, ε) and uτ ∈ D(τ ),
where τ (t, ε) = min{τ , τ }. Hence, we get Rn (|U(t,τ )uτ |≥M )
U(t, τ )uτ p dx
=
Rn (|U(t,τ )uτ |≥M )
|u| – M + M
≤
p– Rn (|U(t,τ )uτ |≥M )
≤ p–
Rn (|U(t,τ )uτ |≥M )
p
dx
p |u| – M dx +
|u| – M
p
dx +
p
Rn (|U(t,τ )uτ |≥M )
Rn (|U(t,τ )uτ |≥M )
Finally, we obtain (.) and the proof is complete.
(M ) dx
|u| – M
p
dx ≤ p ε.
By Theorem ., Lemma . and Lemma ., we can obtain that the process {U(t, τ )}τ ≤t associated with the initial value problem (.) is D-pullback ω-limit compact in Lp (Rn ). So it is easy to prove the following theorem. Theorem . Assume that (.)-(.) hold and the external force g ∈ Lloc (R, L (Rn )) satisfies (.). Then the family of sets A = {A (t) : t ∈ R} is (L (Rn ), Lp (Rn ))-D-pullback attractors for {U(t, τ )}τ ≤t .
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Proof We know that the family Bˆ = {B (t) : t ∈ R} is (L (Rn ), Lp (Rn ))-D-pullback absorbing for the process {U(t, τ )}τ ≤t , where B (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. Consent for publication Not applicable. Authors’ contributions The author read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 7 August 2017 Accepted: 25 September 2017 References 1. Chepyzhov, VV, Vishik, MI: Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl. 73, 279-333 (1994) 2. Song, HT, Zhong, CK: Attractors of non-autonomous reaction-diffusion equations in Lp . Nonlinear Anal. 68, 1890-1897 (2008) 3. Ma, QF, Wang, SH, Zhong, CK: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J. 51, 1541-1559 (2002) 4. Zhong, CK, Yang, MH, Sun, CY: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J. Differ. Equ. 223, 367-399 (2006) 5. Wang, BX: Attractors for reaction-diffusion equations in unbounded domains. Physica D 128, 41-52 (1999) 6. Robinson, JC: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001) 7. Temam, R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997) 8. Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics. American Mathematical Society Colloquium Publications, vol. 49. Am. Math. Soc., Providence (2002) 9. Crauel, H, Flandoli, F: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365-393 (1994) 10. Caraballo, T, Lukaszewicz, G, Real, J: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. 64, 484-498 (2006) 11. Crauel, H, Debussche, A, Flandoli, F: Random attractors. J. Dyn. Differ. Equ. 9, 307-341 (1995) 12. Kloeden, PE, Schmalfuss, B: Asymptotic behaviour of nonautonomous difference inclusions. Syst. Control Lett. 33, 275-280 (1998) 13. Wang, YH, Wang, LZ, Zhao, WJ: Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains. J. Math. Anal. Appl. 336, 330-347 (2007) 14. Li, YJ, Zhong, CK: Pullback attractors for the norm-to-weak continuous process and application to the non-autonomous reaction-diffusion equations. Appl. Math. Comput. 190, 1020-1029 (2007) 15. Zhang, YH, Zhong, CK, Wang, SY: Attractors in L2 (Rn ) for a class of reaction-diffusion. Nonlinear Anal. 71, 1901-1908 (2009) 16. Zhang, YH, Zhong, CK, Wang, SY: Attractors in Lp (Rn ) and H1 (Rn ) for a class of reaction-diffusion. Nonlinear Anal. 72, 2228-2237 (2010) 17. Li, YJ, Wang, SY, Zhong, CK: Pullback attractors for non-autonomous reaction-diffusion equations in Lp . Appl. Math. Comput. 207, 373-379 (2009) 18. Yan, XJ, Zhong, CK: Lp -Uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains. J. Math. Phys. 49, 102705 (2008) 19. Park, SH, Park, JY: Pullback attractor for a non-autonomous modified Swift-Hohenberg equation. Comput. Math. Appl. 67, 542-548 (2014) 20. Yang, L, Yang, MH, Kloeden, PE: Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete Contin. Dyn. Syst. 17, 2635-2651 (2012) 21. Deimling, K: Nonlinear Functional Analysis. Springer, New York (1995)
