The well-posedness and upper semicontinuity of pullback attractors are established for the problem without the uniqueness of solutions under singular perturbations

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 208, pp. 1–15.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

WELL-POSEDNESS OF NON-AUTONOMOUS DEGENERATE PARABOLIC EQUATIONS UNDER SINGULAR

PERTURBATIONS

JINGYU WANG, YEJUAN WANG, DUN ZHAO

Abstract. This article concerns the asymptotic behavior of the following non- autonomous degenerate parabolic equation with singular perturbations defined on a bounded domain inRⁿ,

∂u

∂t +λu−div(|∇u|^p−2∇u)−εdiv“˛

˛∇∂u

∂t

˛

˛

p−2∇∂u

∂t

”

+f(x, t, u) =g(x, t),

whereλis a positive constant,p >2 andε∈(0,1]. The well-posedness and upper semicontinuity of pullback attractors are established for the problem without the uniqueness of solutions under singular perturbations.

1. Introduction

Let Ω be a bounded domain inRⁿ with sufficiently regular boundary∂Ω. Con- sider the following non-autonomous degenerate parabolic equation under singular perturbations defined in Ω fort > τ withτ∈R,

∂u

∂t +λu−div(|∇u|^p−2∇u)−εdiv

|∇∂u

∂t|^p−2∇∂u

∂t

+f(x, t, u) =g(x, t), (1.1) with boundary condition

u(x, t) = 0, x∈∂Ω andt > τ, (1.2) and initial condition

u(x, τ) =uτ(x), x∈Ω, (1.3)

whereλ >0 andp >2 are constants andε∈(0,1].

Nonclassical diffusion equations have been used to model physical phenomena, for instance non-Newtonian flows, soil mechanics, heat conduction, etc (see, e.g., [1, 12, 17]). In the case of p = 2, the upper semicontinuity of global attractors of (1.1)–(1.3) has been studied by several authors in [2, 3, 20, 21, 24] and the references therein as well as [5] for some interesting results on the attractors for delay systems. The stability result of pullback attractors for multi-valued processes was established in [23], and the upper semicontinuity of pullback attractors for nonclassical diffusion equations without the uniqueness of solutions under singular perturbations was addressed.

2010Mathematics Subject Classification. 34K26, 35B41, 35K65.

Key words and phrases. Pullback attractor; multi-valued process; upper semicontinuity;

p-laplacian equation; singular perturbation.

c

2016 Texas State University.

Submitted August 3, 2015. Published August 2, 2016.

1

Recently, the existence and upper semicontinuity of pullback attractors have been proved in [15] for multi-valued processes generated by non-autonomous lat- tice nonclassical diffusion delay systems, in particular, the operator (−1)^p4^p = (−1)^p4 ◦ · · · ◦ 4,ptimes, has been considered instead of−4, wherepis any posi- tive integer and4denotes the discrete one-dimensional Laplace operator. For the continuous case, thep-Laplace operator was defined as

∆pu= div(|∇u|^p−2∇u)

=|∇u|^p−4

|∇u|²4u+ (p−2)

n

X

i,j=1

∂u

∂xi

∂u

∂xj

∂²u

∂xi∂xj

.

The well-posedness and continuity of attractors forp-laplacian problems have been investigated in [14] when the diffusion increases to infinity. For the non-autonomous equation, the existence of uniform attractors and pullback attractors of non-au- tonomous degenerate parabolic equations have been proved in [7, 19]. The existence of random attractors for p-Laplace equations driven by deterministic and stochas- tic forcing was studied in [18], in addition, the upper semicontinuity of random attractors was presented as the intensity of noise approaches zero.

In this article, we assume that the nonlinearityf ∈ C(Ω×R×R;R) and the external forceg satisfy the following conditions:

(H1) the functionF(x, t, s) =Rs

0 f(x, t, ω)dω satisfies

F(x, t, s)>γ₁|s|^q−ϕ₁(x), (1.4)

|F_t⁰(x, t, s)|6α₀F(x, t, s) +ϕ₂(x, t), (1.5) whereγ1>0 andq>2 are constants,α0is sufficiently small, the functions ϕ1∈L¹(Ω) andϕ2∈L¹_loc(R;L¹(Ω)) satisfies

Z t

−∞

Z

Ω

e^αr|ϕ2(x, r)|dx dt <∞, ∀t∈R;

(H2) there exist positive constants γ2, γ3 and functions ϕ3 ∈ L¹_loc(R;L¹(Ω)), ϕ4∈L^q1(Ω) such that

f(x, t, s)s>γ2F(x, t, s)−ϕ3(x, t), (1.6)

|f(x, t, s)|6γ3|s|^q−1+ϕ4(x), (1.7) Z t

−∞

Z

Ω

e^αr|ϕ3(x, r)|dx dt <∞, ∀t∈R, where ¹_q +_q¹

1 = 1;

(H3) the external force g∈L²_loc(R;L²(Ω)) satisfies Z t

−∞

Z

Ω

e^αr|g(x, r)|²dx dt <∞, ∀t∈R, (1.8) whereαis a fixed number given in Lemma 3.2.

The main goal of this paper is to establish the well-posedness and upper semicon- tinuity of pullback attractors for (1.1)–(1.3) under singular perturbations. Because of the lack of the uniqueness of solutions, in order to obtain the pullback attrac- tor we use the general theory of attractors for multi-valued processes developed in [4, 22]. Comparing with the case ofp= 2 the main new difficulty which appears is to deal with the forth term in (1.1).

This article is organized as follows. In Section 2, we recall basic concepts and some necessary results concerning multi-valued processes and pullback attractors.

Section 3 is devoted to the asymptotic behavior of (1.1)–(1.3). The well-posedness and upper semicontinuity of pullback attractors for (1.1)–(1.3) under singular per- turbations are given in Section 4.

The following notation will be used throughout the paper. LetH =L²(Ω) with the normk · k2 and inner product (·,·), and letV =W₀^1,p(Ω). The norm ofL^p(Ω) is written ask · kp. The letter C is a generic positive constant which may change their values from line to line or even in the same line.

2. Preliminaries

LetX be a Banach space with norm k · kX, and let 2^X be the collection of all subsets ofX. Denote byH_X^∗(·,·) the Hausdorff semidistance between two nonempty subsets of a Banach space (X,k · kX), which are defined by

H_X^∗(A, B) = sup

a∈A

dist_X(a, B),

where distX(a, B) = infb∈Bka−bkX. Finally, denote byN(A, r) the open neigh- borhood{y∈X: dist_X(y, A)< r}of radiusr >0 of a subsetAof a Banach space X.

Definition 2.1. A family of mappingsU(t, τ) :X →2^X, t>τ, τ ∈R,is called to be a multi-valued process (MVP in short) if it satisfies:

(1) U(τ, τ)x={x} for allτ ∈R,;x∈X;

(2) U(t, s)U(s, τ)x=U(t, τ)xfor allt>s>τ,τ∈R, x∈X.

LetD be a nonempty class of parameterized setsD={D(t)}t∈R⊂2^X.

Definition 2.2. A collectionD of some families of nonempty subsets ofX is said to be inclusion-closed if for eachD ∈D,

{D(t) : ˜˜ D(t) is a nonempty subset ofD(t),∀t∈R} (2.1) also belongs toD, see, e.g., [9].

Definition 2.3. Let{U(t, τ)} be a multi-valued process onX.

(1)Q={Q(t)}_t∈R∈D is called aD-pullback absorbing set for{U(t, τ)} if for anyB={B(t)}_t∈R∈D and eacht∈R, there exists at0=t0(B, t)∈R⁺such that

U(t, t−s)B(t−s)⊂Q(t), ∀s>t0.

(2) {U(t, τ)} is said to beD-pullback asymptotically upper-semicompact inX with respect to B if for any fixedt ∈R, any sequence yn ∈ U(t, t−sn)xn has a convergent subsequence in X wheneversn →+∞(n→ ∞),xn ∈B(t−sn) with B={B(t)}_t∈R∈D.

Definition 2.4. A family of nonempty compact subsets A = {A(t)}t∈R ∈ D is called to be a D-pullback attractor for the multi-valued process {U(t, τ)}, if it satisfies

(1) A={A(t)}_t∈Ris invariant, i.e.,

U(t, τ)A(τ) =A(t), ∀t>τ, τ ∈R;

(2) Aattracts every member of D; that is, for everyB={B(t)}t∈R ∈D and any fixedt∈R,

s→+∞lim H_X^∗(U(t, t−s)B(t−s), A(t)) = 0.

Definition 2.5. A mapping ψ : R → X is called a complete orbit of the multi- valued process{U(t, τ)} if for everyτ∈Randt>τ, the following holds:

ψ(t)∈U(t, τ)ψ(τ).

If, in addition, there existsD={D(t)}_t∈R∈D such that ψ(t) belongs toD(t) for everyt∈R, thenψis called aD-complete orbit of{U(t, τ)}.

The following existence result of pullback attractors for multi-valued processes can be found in [4, 22].

Theorem 2.6. LetD be a inclusion-closed collection of some families of nonempty subsets of X and {U(t, τ)} be a multi-valued process on X. Also U has closed values and let U(t, τ)x is norm-to-weak upper-semicontinuous in x for fixed t >

τ, τ ∈ R (i.e., if x_n → x in X, then for any y_n ∈ U(t, τ)x_n, there exists a y ∈ U(t, τ)x such that y_n * y (weak convergence)). Suppose that {U(t, τ)} is D-pullback asymptotically upper-semicompact inX and{U(t, τ)}has aD-pullback absorbing setQ={Q(t)}_t∈R inD. Then, theD-pullback attractor A={A(t)}_t∈R is unique and is given by, for each t∈R,

A(t) =∩_T>0∪_s>TU(t, t−s)Q(t−s)

={ψ(t) :ψ is aD-complete orbit of{U(t, τ)}}. (2.2) Let X be a reflexive and separable Banach space, and let X_w be the spaceX endowed with the weak topology. Since bounded closed and convex subsets in the strong topology are compact in the weak topology (due to Mazur’s lemma), the D-pullback absorbing setQ={Q(t)}_t∈Robtained through ultimately boundedness is compact inX_w. Then in the same way as in Theorem 2.6 we have the following result needed to multi-valued processes without further compactness assumptions.

Theorem 2.7. LetD be a inclusion-closed collection of some families of nonempty subsets ofX and{U(t, τ)}be a multi-valued process onX. Also for any fixedt>τ, τ ∈R, U has weakly closed values and U(t, τ) is weakly upper-semicontinuous in bounded sets. Assume that{U(t, τ)}has aD-pullback absorbing setQ={Q(t)}_t∈R in D, and for all t ∈ R, Q(t) is a weakly closed nonempty subset of X. Then {U(t, τ)} has a uniqueD-pullback attractor Aw={Aw(t)}t∈Rwith weakly compact component sets determined by

Aw(t) =∩T>0∪s>TU(t, t−s)Q(t−s)^w for eacht∈R. (2.3) Note that the component subsetsAw(t) of the pullback attractorAware weakly compact inX, hence they are closed and bounded in the strong norm topology.

3. Existence of solutions and their long time behavior

In this section, we firstly establish the existence of solutions for (1.1)–(1.3), and then give uniform estimates of solutions which are useful for obtaining the existence of pullback attractors.

Theorem 3.1. Suppose(H1), (H2)hold and letg∈L²_loc(R;L²(Ω)). Then for any fixed ε∈(0,1], every τ ∈R and any uτ ∈V ∩L^q(Ω), there exists a solution u(t) to problem (1.1)–(1.3), andu(t)satisfies

u∈C([τ, T];V)∩L^∞(τ, T;L^q(Ω)), ∀T > τ.

Proof. We divide the proof into two steps.

Step 1. Multiplying (1.1) byu+^∂u_∂t and then integrating on Ω, we obtain d

dt

(1 2 +λ

2)kuk²₂+1

pk∇uk^p_p +k∂u

∂tk²₂+λkuk²₂+k∇uk^p_p+εk∇∂u

∂tk^p_p +ε

Z

Ω

|∇∂u

∂t|^p−2∇∂u

∂t∇udx+ Z

Ω

f(x, t, u)(u+∂u

∂t)dx

= Z

Ω

g(x, t)(u+∂u

∂t)dx.

(3.1)

It follows from (1.5) and (1.6) that Z

Ω

f(x, t, u)(u+∂u

∂t)dx

> d dt

Z

Ω

F(x, t, u)dx+ (γ2−α0) Z

Ω

F(x, t, u)dx− kϕ2(t)k1− kϕ3(t)k1.

(3.2)

By (3.2) and Young’s inequality, we deduce from (3.1) that d

dt((1 2 +λ

2)kuk²₂+1

pk∇uk^p_p+ Z

Ω

F(x, t, u)dx) +λ 2kuk²₂ + (1−ε

2)k∇uk^p_p+ε 2k∇∂u

∂tk^p_p+ (γ₂−α₀) Z

Ω

F(x, t, u)dx+1 2k∂u

∂tk²₂ 6Ckg(t)k²₂+kϕ2(t)k1+kϕ3(t)k1.

(3.3)

We chooseα₀sufficiently small such thatα₀< γ₂. Using (1.4), we have d

dt

(1 2+λ

2)kuk²₂+1

pk∇uk^p_p+ Z

Ω

F(x, t, u)dx +λ

2kuk²₂ + (1−ε

2)k∇uk^p_p+1 2k∂u

∂tk²₂+γ1(γ2−α0)kuk^q_q+ε 2k∇∂u

∂tk^p_p 6(γ₂−α₀)kϕ₁k₁+kϕ₂(t)k₁+kϕ₃(t)k₁+Ckg(t)k²₂.

(3.4)

By (1.4), (1.7) and Young’ inequality, it yields Z

Ω

F(x, t, u(t))dx>γ₁ku(t)k^q_q− kϕ₁k₁, (3.5) and

Z

Ω

F(x, τ, u(τ))dx= Z

Ω

Z u(τ)

0

f(x, t, ω)dωdx 6

Z

Ω

Z u(τ)

0

(γ₃|ω|^q−1+ϕ₄(x))dωdx 6Cku(τ)k^q_q+Ckϕ₄k^q_q¹

1.

(3.6)

Then, integrating (3.4) fromτ tot, in view of (3.5) and (3.6), we obtain (1

2 +λ

2)ku(t)k²₂+1

pk∇u(t)k^p_p+γ₁ku(t)k^q_q+λ 2

Z t

τ

ku(r)k²₂dr +1

2 Z t

τ

k∂u(r)

∂r k²₂dr+ (1−ε 2)

Z t

τ

k∇u(r)k^p_pdr +γ1(γ2−α0)

Z t

τ

ku(r)k^q_qdr+ε 2

Z t

τ

k∇∂u(r)

∂r k^p_pdr 6(1

2 +λ

2)ku(τ)k²₂+1

pk∇u(τ)k^p_p+Cku(τ)k^q_q+ Z t

τ

kϕ₂(r)k₁dr +Ckϕ4k^q_q¹₁+kϕ1k1+ (γ2−α0)kϕ1k1(t−τ) +

Z t

τ

kϕ3(r)k1dr +C

Z t

τ

kg(r)k²₂dr.

(3.7)

Step 2. LetA:V →V^∗ be the operator defined by (A(u₁), u₂)_(V∗,V)=

Z

Ω

|∇u₁|^p−2∇u₁· ∇u₂dx, for allu₁, u₂∈V, (3.8) where (·,·)_(V^∗_,V) is the duality pairing of V^∗ and V. Note that A is a monotone operator as in [13]. Let {ej}^∞_j=1 ⊆V ∩L^q(Ω) be an orthonormal basis ofH such that the span{ej :j∈N}is dense inV ∩L^q(Ω). Givenn∈N, letX_n be the space spanned by{ej :j= 1, . . . , n}andP_n:H →X_n be the projection given by

Pnu=

n

X

j=1

(u, ej)ej, ∀u∈H.

Note thatP_n can be extended toV^∗ and (L^q(Ω))^∗ by Pnφ=

n

X

j=1

(φ(ej))ej, forφ∈V^∗orφ∈(L^q(Ω))^∗. Consider the following system foruⁿ ∈Xn defined fort > τ:

duⁿ

dt +λuⁿ+PnA(uⁿ)−εPnA(duⁿ

dt ) +Pnf(·, t, uⁿ) =Png(·, t), (3.9) with initial condition

uⁿ(τ) =P_nu_τ. (3.10)

Then it follows from (3.7) that for anyT > τ,

{uⁿ}^∞_n=1 is bounded in L^∞(τ, T;V)∩L^∞(τ, T;L^q(Ω)), (3.11) {duⁿ

dt }^∞_n=1 is bounded in L^p(τ, T;V). (3.12) Analogous to the proof of [8, Theorem 3.1, Section XV.3] and the argument in [16, Section IV4.4], by a standard argument we obtain that for any fixed ε ∈ (0,1], every τ ∈ R and any u_τ ∈ V ∩L^q(Ω), system (1.1)–(1.3) has a solution u ∈ C([τ, T];V)∩L^∞(τ, T;L^q(Ω)) for anyT > τ.

Based on Theorem 3.1, we can define a family of multi-valued mappingsU^ε(t, τ) : V ∩L^q(Ω)→V ∩L^q(Ω) for eachε >0 by setting

U^ε(t, τ)uτ=

u(t) :u(·) is a solution of (1.1)–(1.3) with uτ ∈V ∩L^q(Ω) . Then we can verify that{U^ε(t, τ)}be a multi-valued process onV ∩L^q(Ω).

LetS be a nonempty bounded subset of the Banach spaceX, and letkSkX = sup_u∈SkukX. We consider a familyD={D(t)}_t∈R of bounded nonempty subsets ofV ∩L^q(Ω) such that for everyt∈R,

s→−∞lim e^αs(kD(t+s)k²_H+kD(t+s)k^p_V +kD(t+s)k^q_Lq(Ω)) = 0, (3.13)

whereα >0 will be given in the proof of Lemma 3.2. In the sequel, we will useD to denote the collection of all families with property (3.13):

D={D={D(t)}_t∈R:Dsatisfies (3.13)}.

It is obvious thatD is inclusion-closed.

To consider the asymptotic behavior of problem (1.1)–(1.3), we need the following uniform estimates of solutions.

Lemma 3.2. Suppose (H1)–(H3) hold. Then the multi-valued process {U^ε(t, τ)}

corresponding to problem (1.1)–(1.3) possesses a closed uniformly (with respect to ε∈(0,1]) D-pullback absorbing set Q={Q(t)}t∈R in D, i.e., for each t∈Rand any B ={B(t)}t∈R ∈D, there exists T =T(B, t) >0 which is independent of ε such that for all ε∈(0,1],

U^ε(t, t−s)B(t−s)⊆Q(t), ∀s>T.

Proof. We chooseαandα0 sufficiently small, such that λ

2kuk²₂+ (1−ε

2)k∇uk^p_p+ (γ2−α0) Z

Ω

F(x, t, u)dx

>α (1

2+λ

2)kuk²₂+1

pk∇uk^p_p+ Z

Ω

F(x, t, u)dx .

Then it follows from (3.3) that d

dt

(1 2 +λ

2)kuk²₂+1

pk∇uk^p_p+ Z

Ω

F(x, t, u)dx +1

2k∂u

∂tk²₂ +α

(1 2 +λ

2)kuk²₂+1

pk∇uk^p_p+ Z

Ω

F(x, t, u)dx +ε

2k∇∂u

∂tk^p_p 6Ckg(t)k²₂+kϕ₂(t)k₁+kϕ₃(t)k₁.

(3.14)

Using Gronwall’s lemma, we deduce that (1

2 +λ

2)ku(t)k²₂+1

pk∇u(t)k^p_p+ Z

Ω

F(x, t, u(t))dx

+1 2e^−αt

Z t

t−s

e^αrk∂u(r)

∂r k²₂dr+ε 2e^−αt

Z t

t−s

e^αrk∇∂u(r)

∂r k^p_pdr 6(1

2 +λ

2)e^−αsku(t−s)k²₂+1

pe^−αsk∇u(t−s)k^p_p +e^−αs

Z

Ω

F(x, t−s, u(t−s))dx+Ce^−αt Z t

t−s

e^αrkg(r)k²₂dr +e^−αt

Z t

t−s

e^αr(kϕ2(r)k1+kϕ3(r)k1)dr.

(3.15)

By a similar arguments as in (3.5) and (3.6), we have (1

2 +λ

2)ku(t)k²₂+1

pk∇u(t)k^p_p+γ₁ku(t)k^q_q +1

2e^−αt Z t

t−s

e^αrk∂u(r)

∂r k²₂dr+ε 2e^−αt

Z t

t−s

e^αrk∇∂u(r)

∂r k^p_pdr 6Ce^−αs(ku(t−s)k²₂+k∇u(t−s)k^p_p+ku(t−s)k^q_q) +Ce^−αskϕ4k^q_q¹

1

+Ckϕ₁k₁+Ce^−αt Z t

t−s

e^αrkg(r)k²₂dr +Ce^−αt

Z t

t−s

e^αr(kϕ₂(r)k₁+kϕ₃(r)k₁)dr,

(3.16)

whereCis independent ofε∈(0,1]. Denote byR(t) the nonnegative number given for eacht∈Rby

(R(t))²=C+Ce^−αt Z t

−∞

e^αrkg(r)k²₂dr, (3.17) and consider the family of closed bounded ballsQ={Q(t)}_t∈RinV∩L^q(Ω) defined by

Q(t) ={ψ∈V ∩L^q(Ω) :kψk²₂+k∇ψk^p_p+kψk^q_q 6(R(t))²}. (3.18) It is straightforward to check that Q ∈ D, and moreover, by (3.13) and (3.16), the family ofQ is uniformly (with respect toε∈(0,1]) D-pullback absorbing for the family of multi-valued processes {U^ε(t, τ)}, ε ∈ (0,1] and thus the proof is

complete.

We recall thatXwbe the Banach spaceX endowed with the weak topology. We say thatuⁿ→u∈C([τ, T];Xw) in C([τ, T];Xw) if

uⁿ(sⁿ)→u(s) in Xw for allsⁿ →s∈[τ, T].

Lemma 3.3. Let {uⁿ_τ}^∞_n=1 be a bounded subset ofV ∩L^q(Ω),uτ ∈V ∩L^q(Ω)and let uⁿ_τ → uτ weakly in V ∩L^q(Ω) as n → ∞. Suppose (H1)–(H3) hold and fix T > τ. Then for any fixed ε∈ (0,1] and any sequence uⁿ(t) ∈U^ε(t, τ)uⁿ_τ, there exist u(t)∈U^ε(t, τ)uτ and a subsequence {u^nk}^∞_k=1 satisfying

u^nk →u weakly inC([τ, T];V), u^nk→u weak-star in L^∞(τ, T;L^q(Ω)).

Proof. Inequality (3.7) implies that

{uⁿ}^∞_n=1 is bounded in L^∞(τ, T;V)∩L^∞(τ, T;L^q(Ω)). (3.19) Hence, there exist a functionu∈L^∞(τ, T;V)∩L^∞(τ, T;L^q(Ω)) and a subsequence {uⁿ}^∞_n=1 (relabeled as{uⁿ}^∞_n=1) such that

uⁿ→u weak-star inL^∞(τ, T;V)∩L^∞(τ, T;L^q(Ω)). (3.20) On the other hand, integrating (3.4) froms1to s2withs1, s2∈[τ, T] ands1< s2, then it follows from the similar argument of (3.7) that

ε Z s₂

s₁

k∇∂uⁿ(r)

∂r k^p_pdr 6(1 +λ)kuⁿ(s1)k²₂+2

pk∇uⁿ(s1)k^p_p+Ckuⁿ(s1)k^q_q+ 2 Z s₂

s1

kϕ2(r)k1dr +Ckϕ4k^q_q¹₁+ 2kϕ1k1+ 2(γ2−α0)kϕ1k1(s2−s1) + 2

Z s₂

s1

kϕ3(r)k1dr +C

Z s2

s₁

kg(r)k²₂dr 6(1 +λ)kuⁿ(τ)k²₂+2

pk∇uⁿ(τ)k^p_p+Ckuⁿ(τ)k^q_q + 4

Z T

τ

kϕ2(r)k1dr+Ckϕ4k^q_q¹₁+ 4kϕ1k1+ 4(γ2−α0)kϕ1k1(T−τ) + 4

Z T

τ

kϕ3(r)k1dr+C Z T

τ

kg(r)k²₂dr,

(3.21)

and thus by H¨older’s inequality, we have k∇uⁿ(s2)− ∇uⁿ(s1)k^p_p6

Z

Ω

(s2−s1)^pp1Z s₂ s₁

|∇∂uⁿ(r)

∂r |^pdr dx

= (s2−s1)^p−1 Z s2

s1

k∇∂uⁿ(r)

∂r k^p_pdr

6 C

ε(s2−s1)^p−1,

(3.22)

where ¹_p + _p¹

1 = 1. From (3.19) we deduce that for any t ∈ [τ, T], the sequence {uⁿ(t)}^∞_n=1 is relatively weakly compact in V ∩L^q(Ω). Arguing as in the proof of [6, Theorem 4], by the diagonal method and (3.22) we obtain the existence of a continuous function v(·) and a subsequence of {uⁿ}^∞_n=1 (denoted again {uⁿ}^∞_n=1) such that

uⁿ→v weakly inC([τ, T];V). (3.23) By the similar argument of the existence of solutions in Theorem 3.1, in view of (3.20) and (3.23), we conclude that u=v is a solution of (1.1)–(1.3) and u(τ) =

v(τ) =uτ, which completes the proof.

Lemma 3.3 implies that for any fixedt>τ,τ ∈R,U has weakly closed values and U(t, τ) is weakly upper-semicontinuous in bounded sets. Thanks to Theorem 2.7 and Lemma 3.2, we obtain the following result.

Theorem 3.4. Suppose(H1)–(H3) hold. Then for anyε∈(0,1], the multi-valued process {U^ε(t, τ)} associated to problem (1.1)–(1.3)possesses a unique D-pullback attractorA^ε_w={A^ε_w(t)}t∈R, which is invariant and pullback attracts every member ofD in the weak topology ofV∩L^q(Ω), and its component sets are weakly compact inV ∩L^q(Ω).

Hence, the component sets are compact in the topology ofH, and A^ε_w pullback attracts every member ofD in the topology ofH.

4. Limit problem and convergence properties

In this section, we study the asymptotic dynamic of the problem (1.1)–(1.3) as ε→0. Whenε= 0, we need to consider the following system defined in Ω fort > τ withτ∈R,

∂u

∂t +λu−div(|∇u|^p−2∇u) +f(x, t, u) =g(x, t), (4.1) with boundary condition

u(x, t) = 0, x∈∂Ω andt > τ, (4.2) and initial condition

u(x, τ) =uτ(x), x∈Ω. (4.3)

Analogous to the arguments in [19] and Theorem 3.4, we obtain the existence of solutions and pullback attractors for (4.1)–(4.3).

Theorem 4.1. Suppose(H1)–(H3)hold and letg∈L²_loc(R;L²(Ω)). Then for every τ∈Rand anyu_τ ∈H, there exists a solutionu(t)to problem(4.1)–(4.3), andu(t) satisfies

u∈C([τ, T];H)∩L^p(τ, T;V)∩L^q(τ, T;L^q(Ω)), ∀T > τ.

We consider a familyD ={D(t)}_t∈R of bounded nonempty subsets of H such that for everyt∈R,

s→−∞lim e^αskD(t+s)k²_H = 0, (4.4) and we will useDH to denote the collection of all families with property (4.4):

DH ={D={D(t)}t∈R:Dsatisfies (4.4)}.

It is clear thatDH is inclusion-closed.

Theorem 4.2. Suppose(H1)–(H3) hold. Then

(1) there exists a uniqueDH-pullback attractor A⁰={A⁰(t)}_t∈Rfor the multi- valued process{U⁰(t, τ)} onH generated by problem (4.1)–(4.3);

(2) the multi-valued process {U⁰(t, τ)} possesses pullback attractors A⁰_V,w = {A⁰_V,w(t)}_t∈RandA⁰_Lq,w ={A⁰_Lq,w(t)}_t∈Rin the weak topology, their compo- nent sets are weakly compact inV andL^q(Ω)and hence closed and bounded in the topology of V and L^q(Ω), A⁰_V,w and A⁰_Lq,w pullback attract every member ofDH in the weak topology ofV andL^q(Ω), respectively.

Now we present the equi-continuity of solutions of problem (1.1)–(1.3), which will be used in the proof of Theorem 4.4.

Lemma 4.3. Suppose (H1)–(H3) hold. Then for any fixed T > τ, every uτ ∈ V ∩L^q(Ω) and any s1, s2 ∈ [τ, T] with s1 < s2, any solution u of (1.1)–(1.3) satisfies

ku(s2)−u(s1)k26C(s2−s1)^1/2, whereC is independent of ε∈(0,1].

Proof. Integrating (3.4) from s₁ to s₂ with s₁, s₂ ∈ [τ, T] and s₁ < s₂, then it follows from the similar argument of (3.7) that

Z s₂

s1

k∂u(r)

∂r k²₂dr 6(1 +λ)ku(s1)k²₂+2

pk∇u(s1)k^p_p+Cku(s1)k^q_q+ 2 Z s2

s1

kϕ2(r)k1dr +Ckϕ4k^q_q¹₁+ 2kϕ1k1+ 2(γ₂−α₀)kϕ1k1(s₂−s₁) + 2

Z s2

s₁

kϕ3(r)k1dr +C

Z s₂

s₁

kg(r)k²₂dr 6(1 +λ)ku(τ)k²₂+2

pk∇u(τ)k^p_p+Cku(τ)k^q_q+ 4 Z T

τ

kϕ2(r)k1dr +Ckϕ₄k^q_q¹

1+ 4kϕ₁k₁+ 4(γ₂−α₀)kϕ₁k₁(T−τ) + 4

Z T

τ

kϕ₃(r)k₁dr+C Z T

τ

kg(r)k²₂dr,

(4.5)

and by H¨older’s inequality, we have ku(s2)−u(s1)k²₂6

Z

Ω

(s2−s1)Z s₂ s₁

|∂u(r)

∂r |²dr dx

= (s2−s1) Z s2

s1

k∂u(r)

∂r k²₂dr.

(4.6)

Then the conclusion follows immediately from (4.5) and (4.6).

Theorem 4.4. Suppose(H1)–(H3)hold, let{u^ε_τ :ε∈(0,1]}is a bounded subset of V ∩L^q(Ω),u⁰_τ ∈H and let u^ε_τ →u⁰_τ in the topology of H asε→0. Then for any fixed T > τ and any sequence u^ε of (1.1)–(1.3) with initial data u^ε_τ, we can find a solution u⁰ of (4.1)–(4.3)with initial data u⁰_τ and a subsequence of{u^ε} which converges to u⁰ in C([τ, T];H)and weakly tou⁰ in L^p(τ, T;V)∩L^q(τ, T;L^q(Ω)).

Proof. We divide the proof into two steps.

Step 1. Letεn∈(0,1] be a sequence of positive numbers withεn →0 (n→ ∞), and letu^εn be the solution of (1.1)–(1.3) withu^εn(τ) =u^ε_τⁿ. It follows from (3.7) that for anyt∈[τ, T],

{u^εn(t)}^∞_n=1 is bounded inV ∩L^q(Ω), (4.7) {u^εn}^∞_n=1 is bounded in L^∞(τ, T;H)∩L^∞(τ, T;V)∩L^∞(τ, T;L^q(Ω)), (4.8) consequently

{u^εn}^∞_n=1 is bounded inL^p(τ, T;V)∩L^q(τ, T;L^q(Ω)), {∂u^εn

∂t }^∞_n=1 is bounded in L²(τ, T;H), (4.9)

{Au^εn}^∞_n=1 is bounded in L^p1(τ, T;V^∗) with 1 p+ 1

p1

= 1, (4.10) {f(t, x, u^εn)}^∞_n=1is bounded in L^q1(τ, T;L^q1(Ω)) with 1

q+ 1 q1

= 1, (4.11) ε^1/p_n k∇∂u^εn

∂t k_Lp(τ,T;L^p(Ω))6C0, (4.12) for some constant C₀ > 0. Hence, there exist a function u⁰ ∈ L^∞(τ, T;H)∩ L^p(τ, T;V)∩L^q(τ, T;L^q(Ω)) and a subsequence of{u^εn}^∞_n=1(relabeled as{u^εn}^∞_n=1) such that

u^εn→u⁰ weak-star inL^∞(τ, T;H), (4.13) u^εn→u⁰ weakly inL^p(τ, T;V) andL^q(τ, T;L^q(Ω)), (4.14) A(u^εn)→χ1 weakly inL^p1(τ, T;V^∗), (4.15) f(t, x, u^εn)→χ₂ weakly inL^q1(τ, T;L^q1(Ω)). (4.16) Thanks to Lemma 4.3, (4.8) and the compactness of embedding V ,→ H, by the Ascoli-Arzel`a theorem we deduce that there exists a subsequence of{u^εn}^∞_n=1 (de- noted again{u^εn}^∞_n=1) such that

u^εn→u⁰ strongly inC([τ, T];H). (4.17) Step 2. It remains to show that u⁰ is a solution of (4.1)–(4.3) withu⁰(τ) =u⁰_τ. Noticing thatu^εn is a solution of (1.1)–(1.3) withu^εn(τ) =u^ε_τⁿ, i.e., u^εn satisfies

∂u^εn

∂t +λu^εn−div(|∇u^εn|^p−2∇u^εn)

−ε_ndiv(|∇∂u^εn

∂t |^p−2∇∂u^εn

∂t ) +f(x, t, u^εn) =g(x, t),

(4.18) from (4.12) and H¨older’s inequality, we obtain that for anyξ∈V ∩L^q(Ω),

−ε_n Z T

τ

div(|∇∂u^εn(t)

∂t |^p−2∇∂u^εn(t)

∂t ), ξ

(V^∗,V)

dt

6ε_nk∇∂u^εn(t)

∂t k^p−1_Lp(τ,T;L^p(Ω))k∇ξk_Lp(τ,T;L^p(Ω))

6ε^1/p_n C₀^p−1k∇ξkL^p(τ,T;L^p(Ω))→0 asn→ ∞.

(4.19)

By (4.12)–(4.17) and (4.19), one can show that d

dt(u⁰, ξ) +λ(u⁰, ξ) + (χ1, ξ)_(V^∗_,V)+ (χ2, ξ)_(L^q1,L^q)= (g(t), ξ). (4.20) SinceV ,→H is compact, in view of (4.8) and (4.9), up to a subsequence we have u^εn→u⁰ in L²(τ, T;L²(Ω)), (4.21) which implies

u^εn→u⁰ for almost every (t, x)∈[τ, T]×Ω. (4.22) From this and the continuity off, we obtain

f(x, t, u^εn)→f(x, t, u⁰) for almost every (t, x)∈[τ, T]×Ω. (4.23) By (4.16) and (4.23), we have

χ₂=f(x, t, u⁰). (4.24)

Finally, by the similar argument of (4.19), in view of (4.8) and (4.12), we find that

−εn

Z T

τ

div(|∇∂u^εn(t)

∂t |^p−2∇∂u^εn(t)

∂t ), u^εn(t)

(V^∗,V)dt

6εnk∇∂u^εn(t)

∂t k^p−1_Lp(τ,T;L^p(Ω))k∇u^εn(t)kL^p(τ,T;L^p(Ω))

6ε^1/p_n C→0

(4.25)

asn→ ∞. By (4.13)–(4.17) and (4.25), we can argue as in [18] to show that

χ1=A(u⁰). (4.26)

It follows from (4.20), (4.24) and (4.26) thatu⁰is a solution of problem (4.1)–(4.3),

and thus the proof of this theorem is complete.

We obtain the upper semicontinuity of pullback attractors under singular per- turbations.

Theorem 4.5. Suppose (H1)–(H3) hold, and let A^ε_w = {A^ε_w(t)}_t∈R and A⁰ = {A⁰(t)}_t∈R be the pullback attractors for the multi-valued processes {U^ε(t, τ)} and {U⁰(t, τ)} in D and DH generated by (1.1)–(1.3) and (4.1)–(4.3), respectively.

Then for anyτ ∈R,

ε→0limH^∗ A^ε_w(τ), A⁰(τ)

= 0, (4.27)

whereH^∗(·,·) is the Hausdorff semidistance between two nonempty subsets ofH. Proof. We will use the argument of contradiction. Indeed, assume that (4.27) is not true, then there exist a τ ∈ R, a positive constant η, a sequence of positive numbers εn converging to zero, and a corresponding sequence u^ε_τⁿ ∈ A^εn(τ) such that

dist_H(u^ε_τⁿ, A⁰(τ))>η >0, ∀n∈N. (4.28) Letu^εn be a solution of (4.1)–(4.3) with initial condition u^εn(τ) =u^ε_τⁿ. It is clear thatu^εn(t) belongs toA^εn(t) for allt>τ. Note thatA^ε_w={A^ε_w(t)}_t∈Ris invariant, hence there existsu^ε_τ−1ⁿ ∈A^εn(τ−1) such thatu^ε_τⁿ∈U^εn(τ, τ−1)u^ε_τ−1ⁿ . If we now take u^εn(s) ∈U^εn(s, τ −1)u^ε_τ−1ⁿ fors ∈ [τ−1, τ], then we haveu^εn(s) ∈A^εn(s) for all s > τ−1. Applying the above procedure several times we can construct u^εn(s) ∈ A^εn(s) for all s > τ−m, m ∈ N. Letting m → ∞, we obtain a D- complete orbit of u^εn(s), s ∈ R, of the multi-valued process U^εn(t, τ) such that u^εn(s)∈A^εn(s) for alls∈R.

For anyt∈R, since{u^εn(t)}^∞_n=1is a bounded subset ofV∩L^q(Ω), there exists a subsequence of{u^εn(t)}^∞_n=1 (relabeled as{u^εn(t)}^∞_n=1) such thatu^εn(t)→u⁰(t) in H asn→ ∞. Then, using Theorem 4.4 and the diagonal method, one can choose a subsequence of{u^εn(·)} and aDH-complete orbitu⁰of (4.1)–(4.3) such that

u^εn(t)→u⁰(t) inC(J;H) (4.29) for any compact interval J ⊂ R. Theorem 2.6 implies that u⁰(t) ∈ A⁰(t) for all t ∈ R, this and (4.29) lead to a contradiction with (4.28), hence the proof is

completed.

Acknowledgments. The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. It is their contributions that greatly improve the paper. The authors also thank the editors for their kind help. This work was supported by NSF of China (Grant No. 11571153 and 11475073), the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2016-100, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Jingyu Wang

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China.

College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

E-mail address:[email protected]

Yejuan Wang (corresponding author)

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

E-mail address:[email protected]

Dun Zhao

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

E-mail address:[email protected]