Regularity of the attractor for strongly damped wave equations with nonlinearity

R E S E A R C H

Open Access

Regularity of the attractor for strongly

damped wave equations with nonlinearity

Hongyan Li

*_{Correspondence:} [email protected] College of Management, Shanghai University of Engineering Science, Shanghai, 201620, China

Abstract

The long time behavior of the solutions for the strongly damped wave equation is considered with nonlinear damping, a nonlinear forcing term, and with a periodic boundary condition. We prove that the global attractor which captures all trajectories inH1₍

MSC: 35B40; 34A35; 37L30

Keywords: global attractor; strongly damped wave equation; compact set

1 Introduction

Foru(x,t) :×R+→R, we consider the strongly damped wave equation on a bounded domaininR_{with smooth boundary:}

utt–αut–u+h(ut) +f(u) =g(x), (.)

supplemented with the periodic boundary conditions and initial conditions

u(,x) =u(x), ut(,x) =u(x), x∈, (.)

where the strongly damped coefficientαis a positive constant, the damped functionh:

R→Ris continuous, and the forcing termf :R→Ris a nonlinear term satisfying some growth conditions,g:→Ris the external force.

Whenα= , (.) reduces to a usual wave equation with nonlinear damping, which arises as an evolutionary mathematical model in various systems (cf.[, ]), for example: (i) mod-eling a continuous Josephson junction with specifich,gandf; (ii) modeling a hybrid sys-tem of nonlinear waves and nerve conduct; and (iii) modeling a phenomenon in quantum mechanics and which has been studied widely by using of the concept of global attractors; see, for example, [–] for the linear damping case, and [–] for the nonlinear damp-ing case. There are some results on the regularity in these papers; for example, in [] the authors gave the dimensionality and related properties of the global attractor, and in [], the authors presented a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation (whenα=  andh(ut) =ut) with nonlinearity

for the critical growth. But forα= , the case can be complex.

There is a large literature on the asymptotic behavior of solutions to (.), (.) (cf.[, –]). For the strongly damped wave equations, [] had proved the uniform

ness of the global attractor for very strong damping inH_×L_{and obtained an estimate} of the upper bound of the Hausdorff dimension of an attractor for the strongly damped nonlinear wave equation (.). Furthermore, whenh(ut) = , Pata and Zelik in [] had

proved the existence of compact global attractors of optimal regularity,i.e., the compact global attractor onH_()×L_{() is a bounded subset ofH}_×H_{. The aim of this paper} is to prove that the dynamical system associated with (.) possesses a compact global at-tractor inH()×H(),i.e., we will prove that the global attractorAinH()×L() is a compact setAinH()×H(). This implies thatA=A.

The paper is organized as follows. In Section , we present the existence, uniqueness, and continuous dependence of solutions for problem (.), and we establish the existence of an absorbing set inH_()×L_{(). In Section , we prove the existence of the global} attractor inH()×L(). In Section , we establish the regularity of the attractor,i.e., that the global attractor is a compact subset inH_()×H_().

2 Preliminaries

We assume thatg∈L_{() =H, andH}˙–_{= (H}˙₎∗_{, with}

˙

H=H∩

u∈L();

u(x)dx= 

,

and the functionsh,f satisfy the following conditions:

(i) Letf(u)∈C_{(R;R)satisfy:}

() The asymptotic sign condition

lim sup

|s|→+∞

f(s)

s ≤. (.)

() LetF(s) =_sf(ρ)dρ, there exist constantsc,c,δ,Cδ> such that

f(u),u–c

F(u)dx–δ|u|≥–Cδ||, (.)

where| · |denotes the absolute value of the number inR. We have

f(s)≤c

 +|s|p with ≤p<∞, whenn= , ,

≤p< , whenn= . (.)

(ii) There exist two constantsβ_,β_≥such that

h() = ,  <β_≤h(s)≤β_< +∞, ∀s∈R. (.)

LetAu= –u, then the system (.)-(.) is equivalent to the following initial value prob-lem inH˙_()×L_():

˙

Y=PY+Q(Y), x∈,t> ,

whereY= (u,ut)T,Q(Y) = (, –h(ut) –f(u) +g)T,

P=

 I

–A –kA

.

By the assumption (.)-(.), it is easy to check that the functionQ(Y) :H˙×L→ ˙H× L _{is continuously differentiable and globally Lipschitz continuous with respect toY. By} the classical theory concerning the existence and uniqueness of the solutions of evolution differential equations (cf.[]), we have the following lemma (see [] for details).

Lemma . Consider the initial value problem(.)inH˙×L.If(.)-(.)hold,then,for any Y∈ ˙H_×L_{,there exists a unique continuous function Y(·) =Y(·,Y)∈C(R}

+;H˙×L) such that Y(,Y) =Yand Y(t)satisfies the integral equation

Y(t,Y) =ePtY+ t



eP(t–τ)_QY(τ)dτ.

Y(t,Y)is called the mild solution of(.),and Y(t,Y)is jointly continuous in t and Y, and

(u,ut)∈C

R+;H˙()

×CR+;L()

∩L,T∗;H˙(), ∀T∗> .

For anyt≥, we can introduce a map

S(t) :Y= (u,u)→(u,ut) =Y(t,Y), S(t) :H˙×L→ ˙H×L,

whereY(t,Y) is the solution of (.), and then{S(t)|t≥}is a continuous semigroup on ˙

H_×L_.

Consider the mapG:H˙_×L_{→ ˙H}_×L_{defined as}

Gu(t),ut(t)

=ePt(u,u) + t



eP(t–τ)_,g–hu

t(τ)

–fu(τ)dτ. (.)

For someλ> , let

A=A+λI. (.)

Let

(u,v) =

uv dx, |u|= (u,u)_, ∀_u,v∈_L_,

(u,v)=

Au·v dx, u=(u,u) 

_, ∀u_,v∈_H_().

Obviously,Ais symmetric and positive definite, so we have the following Poincaré type inequality:

Leth(ut) =h(ut) –αλut, then (.) can be written in the equivalent form

utt+αAut+Au+h(ut) –λu+f(u) =g, (.)

and by (.),h(ut) satisfies

h() = , –αλ<β≤h(θ)≤β< +∞, ∀θ∈R, (.) whereβ=β–αλ,β=β–αλare two constants.

To construct an attractor for (.), we make the following assumptions. Letϕ= (u,v)T_,

v=ut+ku, wherekis chosen as

k= αλ+β  + (αλ+β)α+β/λ

. (.)

Equation (.) can be written as

ϕt+H(ϕ) =F(ϕ), ϕ() = (u,v=u+ku)T, t≥, (.)

where

F(ϕ) =



λu–f(u) +g

,

H(ϕ) =

ku–v

Au–k(αA–k)u+ (αA–k)v

+

 h(v–ku)

.

We define a new weighted inner product and norm inE=H˙×Las

(ϕ,ψ)E=μ

(u,u)+ (v,v), ϕE= (ϕ,ϕ)/E , (.)

for anyϕ= (u,v)T_{,ψ= (u,v)}T_{∈E, whereμis chosen as}

μ=  + (αλ+β)α+β  /λ  + (αλ+β)α+β/λ

∈

 , 

. (.)

Obviously, the norm · Ein (.) is equivalent to the usual norm| · |H˙_×L inE.

Lemma .(see [], Lemma ) For anyϕ= (u,v)T_∈E,if

–αλ<β≤β< +∞, β≥ |β|+min

/α, (αλ+β)/

, (.)

hold,then

H(ϕ),ϕ_E≥σϕ_E+α v

₊β |v|

_≥σϕ

E+ αλ+β

 |v|

_{, (.)}

where

σ= αλ+β

γ+√γγ

, γ=  + (αλ+β)α+

β_

λ , γ= (αλ+β)α+ β_

Proposition . The semigroup{S(t)|t≥}possesses an absorbing setB⊂ ˙H_×L_.

Proof Letϕ= (u,v)T_{∈Ebe the solution of (.). Taking the inner product (·,·)}

Eof (.)

withϕ, we have

(ϕt,ϕ) +

H(ϕ),ϕ=F(ϕ),ϕ

and

 

d dtϕ



E+

H(ϕ),ϕ_E+f(u),v–λ(u,v) = (g,v). (.)

Byv=ut+ku, we have

f(u),v=f(u),ut+ku

= d

dt

F(u)dx+kf(u),u, (.)

by (.) and (.), we have

 

d dt

ϕ_E+ 

F(u)dx

+H(ϕ),ϕ E+k

f(u),u–λ(u,v) = (g,v). (.)

Lety(t) =ϕ

E+ 

F(u)dx+ Cδ|| ≥



ϕE≥. By (.) and (.), there existsδ> 

such that

H(ϕ),ϕ_E+kf(u),u–λ(u,v)

≥σϕ_E+αλ+β  |v|

_+kc

F(u)dx+kδ|u|–kCδ||–λ|u||v|

≥σϕ_E+αλ+β  |v|

_+kc

F(u)dx

+ kδ

λμμ|u|

_–kC

δ||–

√

λ

√μϕ



E

≥

σ+ kδ

λμ–

√

λ

√μ

ϕ_E+kc

F(u)dx–kCδ||+

αλ+β  |v|



≥ ρy–k

Cδ||+cCδ||

+αλ+β  |v|

_{, (.)}

whereρ=min{kc, (σ+λμkδ –

√

λ

√μ)}. By (.) and (.),

d

dty+ρy≤kCδ||( +c) + (g,v) – (αλ+β)|v| 

≤kCδ||( +c) + 

αλ+β|

g|. (.)

Using the Gronwall lemma, we have

ϕ(t)_E≤y()e–ρt_{+ }

_|g|

(αλ+β)ρ

+kCδ||( +c)

ρ

Following [] and [], it follows from (.) that, for eachκ> , there is a constantCκ> 

such that, for eachu∈L_(),

f(u),u=

f(u)u dx≤κ|u|+Cκ. (.)

Note that (.) and (.) imply

c

F(u)dx≤f(u),u–δ|u|+Cδ||

≤κ|u|+Cκ–δ|u|+Cδ|| ≤c

 +|u|. (.)

For any bounded setBofE, wheresup_ϕ∈BϕE<r, ifϕ()∈B, there existsc=c(r) > 

such thaty() =ϕ()

E+ 

F(u)dx+ Cδ|| ≤c. Therefore, for the solutionϕ(t) =

(u(t),v(t))T_{of (.) withϕ()∈B,}

ϕ(t)_E≤c(r)e–ρt+ 

_|g|

(αλ+β)ρ

+kCδ||( +c)

ρ

.

Taking

M= 

_|g|

(αλ+β)ρ

+kCδ||( +c)

ρ

completes the proof.

3 Existence of the global attractor inH˙1_×L2

Theorem .(see [, I..]) Let{S(t)|t≥}be a continuous semigroup onH˙_×L _that possesses an absorbing ball inH˙_×L_{.Let us assume that,for any t≥,}

S(t) =S(t) +S(t),

where:

• For every bounded setB,there existstthat depends onB,such that_t≥tS(t)Bis relatively compact inH˙_×L_.

• For every bounded setB,

lim sup

t→∞

sup

θ∈B

S(t)θH˙_×L

= .

Then S(t)t≥possesses a global attractorAthat is compact inH˙×L.

Remark .(see []) We characterize such an attractorAas

A=(u,u)∈ ˙H×L,un_,un_n⊂B,∃(tn)→ ∞,

such thatS(tn)

un_,un_– (u,u)_H˙_×L→

.

Proof We considerg∈C∞_ and



g(x)dx=  such that

|g–g|L<, (.)

and we introduce the splitting (u,v) = (u,v) + (u,v) + (u,v) where (u,v) satisfies ⎧

⎪ ⎨ ⎪ ⎩

u,t+ku–v= ,

v,t+Au–k(αA–k)u+ (αA–k)v+h(v–ku) +f(u) –λu=g,

u() = , v() = ,

(.)

(u,v) satisfies

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u,t+ku–v= ,

v,t+Au–k(αA–k)u+ (αA–k)v

+h(v–ku+v–ku) –h(v–ku) +f(u+u) –f(u) –λu=g–g,

u() = , v() = ,

(.)

and (u,v) is the solution of ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u,t+ku–v= ,

v,t+Au–k(αA–k)u+ (αA–k)v

+h(v–ku) –h(v–ku+v–ku) +f(u) –f(u+u) –λu= , u() =u, v() =v.

(.)

We now define the families of maps{S_k(t)|t≥}and{S_k(t)|t≥}inH˙_×L_{, where}

S_k(t)(u,v) =u(t),v(t)+u(t),v(t), S_k(t)(u,v) =u(t),v(t).

First step: We prove that (u,v) is bounded inH×H. The system (.) can be written as

ϕ_t+Hϕ=Fϕ, ϕ() = (, )T, t≥, (.)

where

Fϕ=

 –f(u) +λu+g

,

H

ϕ=

ku–v

Au–k(αA–k)u+ (αA–k)v+h(v–ku)

.

(.)

Similar to Proposition ., we obtain

ϕ_E≤  ρC

|g|H–,λ,α,δ,κ,k,β_,β_

=M. (.)

Now we multiply (.) by (Au,Av) and integrate onto obtain

 

d dtA

  ϕ



E+σA

  ϕ



E+ αλ+β

 A   v



with

–f(u),Av

=–A   f(u),A

 

(u,t+ku)

= –d dt

A

  f(u),A

  u

+A

 

f(u)u,t,A

  u

–kA

  f(u),A

  u

.

Then (.) can be rewritten as

 

d dtA

  ϕ



E+ 

A

  f(u),A

  u

+σA

  ϕ



E+k

A

  f(u),A

  u

≤–αλ+β  A

  v

 +A

 

f(u)u,t,A

  u

+λA

  u,A

  v

+A

  g,A

  v , i.e.,   d dtA

  ϕ



E+ 

A

  f(u),A

  u + σ– √ λ

√μ

A

  ϕ



E+k

A

  f(u),A

  u

≤A  

f(u)u,t,A

  u

+ A

  g

(αλ+β)

; (.)

for the first term on the right-hand side of (.), we have

A

 

f(u)u,t,A

  u

≤cA

 

u,t|u|pA   u 

–p

≤cA  

u,t|u|p_HA  

u–_pA   up_

≤cλp– _A  

u,t|u|p_H|Au| ≤c|Au|_+cλp–_{ A}

u,t|u|_Hp ≤c|Au|+cλ

p–

 _{ +}|_u|

H+|v|_H

≤cA   ϕ



E+cλ

p–

 _{ +}|_u|

H+|v|

. (.)

By (.)-(.), we have

d dtA

  ϕ



E+ 

A

  f(u),A

  u +  σ– √ λ

√μ–c

A

  ϕ



E+ k

A

  f(u),A

  u

≤cλ

p–  |_u|

H+|v|

+ A   g

αλ+β

. (.)

Let=min{σ–

√

λ

√μ–c,k} ≥, using the Gronwall lemma, we have

A   ϕ



E+ 

A

  f(u),A

  u

≤A   ϕ()



E+ 

A   f

u(),A   u()

+ cλp– t



e–tu(s)_H+v(s) 

ds+ |A   g|

(αλ+β)

≤cλ

p– 

t



e–t_u

(s) 

H+v(s) 

ds+ |A   g|

(αλ+β) .

By (.), we have

A   ϕ



E+ 

A

  f(u),A

  u

≤ 

ρC

_A

g,λ,α,δ,κ,k,β,β

.

Note that (.) implies that

A

  f(u),A

  u

≤cA

  u

 +cA

 

u≤cA   ϕ



E.

Then we obtain

A_ϕ

E≤



ρC

_A

g,λ,α,δ,κ,k,β,β

. (.)

Proposition ., (.), and (.) imply that (u,v) is bounded inH_×H_.

Second step: Letϕ= (u,v); we will prove that there exists aK>  independence on

such that

ϕ_E≤K.

Multiplying (.) by (u,v), we thus obtain

 

d dtϕ



E+

σ– √

λ

√_μϕ E+

αλ+β  |v|

_{+f(u+u) –f(u),v}

≤ |g–g||v|. (.)

For the last term on the left-hand side of (.), by (.), Proposition ., and (.), there existsξsuch that

f(u+u) –f(u),v=f(u+ξu)u,v

≥–c +|u|p+|u|p|u||v|

≥–c(M,M)|u||v|. (.)

By the Gronwall and Poincaré inequalities,

ϕ E≤

|g–g|

(

√

λ

√

μ +c(M,M) –σ)(αλ+β)

≤ 

(

√

λ

√_μ +c(M,M) –σ)(αλ+β)

. (.)

By (.), (.), and the following lemma we see that{S

Lemma .(see []) Let X be a complete metric space and A be a subset in X,such that

∀, A⊂K+B

,C()

withlim→C() = and Kis compact in X,then A is compact in X.

Third step: Letϕ_{= (u}_,v_{), by (.) and Lemma ., we have}

 

d dtϕ



E+σϕ



E+ αλ+β

 |v|

_{≤–f(u) –f(u}

+u),v

+λ(u,v). (.)

For the right-hand side of (.), by (.), we have

–f(u) –f(u+u),v+λ(u,v)≤c|u||v|+λ|u||v|

≤cu|v|

≤σ ϕ



E+

c

 σ μ–

σ



|v|_{. (.)}

Letc<σ μ(αλ+β) +σμ. Equations (.) and (.) lead to

ϕ_E≤μu+|v|exp{–σt}. (.)

In other words we have

S(t)(u,v)→ inH˙×L, (.)

uniformly in bounded sets. Then from Theorem ., (.), and the compactness of{S

k(t)|

t≥}, for the system (.) there exists a global attractorAinH˙_×L_.

Remark . It is easy to see that the semigroup

Sk(t) =Sk(t) +Sk(t) : (u,v=u+ku)T→

u(t),ut(t) +ku(t)

T

, E→E,

defined by (.) has the following relation withS(t):

Sk(t) =RkS(t)R–k, (.)

whereRkis an isomorphism ofE:

Rk:{u,ut} → {u,ut+ku}.

Since the semigroup{Sk(t)|t≥}defined by (.) possesses a global attractorA⊂E, by (.),{S(t)|t≥}also possesses a global attractorA=RkA.

4 Regularity of the attractor

In this section, we suppose thatg∈ ˙L_{and prove that the attractor is compact inH}_×H_. Let (u,v)∈ ˙H_×L_{. We set}

Sk(t)(u,v) =

where (p,q) is the solution of ⎧

⎪ ⎨ ⎪ ⎩

pt+kp–q= ,

qt+Ap–k(αA–k)p+ (αA–k)q+h(q–kp) +f(u) –λu=g,

p() = , q() = ,

(.)

(w,ρ) satisfies ⎧ ⎪ ⎨ ⎪ ⎩

wt+kw–ρ= ,

ρt+Aw–k(αA–k)w+ (αA–k)ρ+h(v–ku) –h(q–kp) = ,

w() =u, ρ() =v.

(.)

We use Remark . to prove thatA⊂H_×H_{. Let (u,v)∈A. Let (u}n

,vn)∈Band a sequence of a real numberstn→ ∞asn→ ∞, such that

Sk(tn)

un_,vn_→(u,v) inH˙×Lasn→ ∞. (.)

We also have

Sk(tn)

un_,vn_=pn(tn),qn(tn)

+wn(tn),ρn(tn)

, ∀n∈N. (.)

We deduce from (.) and (.) that

pn(tn),qn(tn)



H_×H≤ 

ρC

_A

g,λ,α,δ,κ,k,β,β

(.)

and

wn(tn),ρn(tn)H˙_×L≤(u,v) 

H_×Lexp(–σtn). (.)

From (.), we infer that there exist subsequencestn and (un

,vn

), and (p,q)∈H×H such that

pn(tn),qn

(tn)

p,q weakly inH×H. (.)

From (.), (.), and (.), we have

pn(tn),qn

(tn)

→(u,v) inH×L. (.)

We conclude that (u,v) = (p_,q_{) and then}

A⊂H×H. (.)

In the following, we use the famous argument of [] to show that the attractorAis actually a compact set inH_×H_.

Proof Multiplying (.) by (Au,Av), we have

 

d dt

μA   u

 +A

  v



+σμA   u

 +A

  v



+αλ+β  A

  v



= (g,Av) –f(u) –λu,Av. (.)

We put G(u,v) = μA

 

u +|A  

v|, K(u,v) =

[A

  g ·A

 

v– (f(u) – λu)Av]dx –

αλ+β  |A

 

v|, then we have

 

d

dtG(u,v) +σG(u,v) =K(u,v), (.)

this shows that

G(u,v) =e–σt_G(u

,v) +  t



eσ(s–t)_{K(u,v)ds. (.)}

We consider a sequence (an,bn)n∈N∈A, and we may assume that, up to a subsequence,

(an,bn)(a,b) weakly inH×H,

(an,bn)→(a,b) strongly inH×L.

We want to prove the strong convergence of (an,bn) inH×H; this will give the

com-pactness ofAinH_×H_.

For a givenT≥ and up to subsequence extraction, we may assume that a.e. int∈[,T],

S(t–T)(an,bn)S(t–T)(a,b) weakly inH×H,

S(t–T)(an,bn)→S(t–T)(a,b) strongly inH×L.

Using (.) for (u,v) =S(–T)(an,bn), we have

GS(t–T)(an,bn)

=e–σtGS(–T)(an,bn)

+  t



eσ(s–t)_KS(s–T)(a

n,bn)

ds. (.)

By the Lebesgue dominated convergence theorem, similar to (.)-(.), we have

t



eσ(s–t)KS(s–T)(an,bn)

ds→

t



eσ(s–t)KS(s–T)(a,b)ds, (.)

passing to the limitsupin (.) and takingt=Twe have

lim

n→∞G(an,bn)≤n→∞lim e

–σT_GS(–T)(a n,bn)

+  T



The last term of (.) follows from (.) for (u,v) =S(–T)(a,b) andt=T, and we have

I=  T



eσ(s–T)KS(s–T)(a,b)ds=G(a,b) –e–σTGS(–T)(a,b). (.)

We replaceIin (.), we obtain

lim

n→∞G(an,bn)≤G(a,b) –e

–σT_{GS(–T)(a,b)– lim} n→∞G

S(–T)(an,bn)

. (.)

It is a standard matter to prove

e–σTGS(–T)(a,b)– lim

n→∞G

S(–T)(an,bn)

→ whenT→ ∞. (.)

It follows from (.) that

lim

n→∞G(an,bn)≤G(a,b)≤_n→∞lim G(an,bn). (.)

This completes the proof of lemma.

Competing interests

The author declares to have no competing interests.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11101265) and Shanghai Education Research and Innovation Key Project (14ZZ157).

Received: 14 March 2014 Accepted: 9 September 2014 Published:16 Oct 2014

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