Bounded solutions for the nonlinear wave equation

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R E S E A R C H

Open Access

Bounded solutions for the nonlinear wave

equation

Tacksun Jung

1

_{and Q-Heung Choi}

2*

*_{Correspondence:} [email protected] 2_{Department of Mathematics}

Education, Inha University, Incheon, 402-751, Korea

Full list of author information is available at the end of the article

Abstract

We investigate the number of periodic weak solutions for the wave equation with nonlinearity decaying at the origin. We get a theorem which shows the existence of a bounded weak solution for this problem. We obtain this result by approaching the variational method and applying the critical point theory for the indeﬁnite functional induced from the invariant subspaces and the invariant functional.

MSC: 35L05; 35L70

Keywords: wave equation; critical point theory; invariant function; invariant subspace; (P.S.)ccondition; eigenvalue problem

1 Introduction and statement of the main result

Letg(x,t,u) be aCfunction from [,π]×R×RtoRandT-periodic int. In this paper we are concerned with the number of weak periodic solutions of the following wave equation with boundary and periodic conditions:

utt–uxx=g(x,t,u),

u(,t) =u(π,t) = ,

u(x,t+T) =u(x,t),

ut(x, ) =ut(x,T) ∀x∈[,π],

(.)

whereT is a rational multiple ofπ. We assume thatgsatisﬁes the following conditions:

(g) g∈C_([,_π]_×_R_×_R,_R)_is_T_{-periodic in}_t_,

(g) g(x,t, ) = ,g(x,t,ξ) =o(|ξ|)uniformly with respect tox∈[,π]andt∈R, (g) there existsC> such that|g(x,t,ξ)|<C∀x∈[,π],t∈R,ξ∈R.

Nonlinear problem of this type has been considered by many authors (cf.[–]).

The purpose of this paper is to show the existence ofT-periodic weak solutions of prob-lem (.).

Our main result is as follows.

Theorem . Assume that g satisﬁes(g)-(g).Then problem(.)has at least one bounded solution.

For the proof of our main result, we approach the variational method and apply the critical point theory induced from the invariant subspaces and the invariant functional.

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The outline of the proof of Theorem . is as follows. In Section , we introduce two Banach spacesHandEof functions satisfying some symmetry properties, stable byA(Au=utt–

uxx),g such that the intersection ofHwith the kernel ofAis reduced to . The search

of a solution of problem (.) in the spaceH reduces the problem to a situation where A–is a compact operator. In Section , we introduce a functionalIdeﬁned onEwhose critical points and weak solutions of (.) possess one-to-one correspondence. We prove thatI∈C_(E,_{R) and satisﬁes the Palais-Smale condition. By a critical point theorem for}

indeﬁnite functionals (cf.[]), we prove that there exists at least one solution of (.) which is bounded, and we prove Theorem ..

2 Invariant Banach space

Let= (,π)×(,T);Tis a rational multiple ofπ

T=πb a ,

whereaandbare coprime integers. LetAbe the operator deﬁned by

D(A) =u∈C()|u(,t) =u(π,t) =  ∀t∈[,T],

u(x, ) =u(x,T) ∀x∈[,π],

ut(x, ) =ut(x,T) ∀x∈[,π]

,

Au=utt–uxx.

LetAbe the adjoint ofAinL_{(). We investigate the weak solutions of}

Au=g(x,t,u).

We note that the eigenvalues ofAareλjk=j– (_Tπk),j= , , . . . andk= , , , . . . , and the

corresponding eigenfunctions are

sinjxsinπkt

T and sinjxcos πkt

T .

We also note that the set of functionssinjxsinπ_Tkt,sinjxcosπ_Tkt is an orthogonal base forL_{(). Let}_u_{be a function of}_L_{(). Then there exists one and only one function of}

L_([,_π_]_×_{R) which is}_T_{-periodic in}_t_{and equals}_u_on_{. We shall denote this function}

byu. Let us denote an elementu, inL_{(), by}

u=

j>,k

ujksinjxexpik

a bt

with

uj,k=u¯j,–k.

We assume thatbis even andais odd. LetHbe the closed subspace ofL_{() deﬁned by}

H=

u∈L()u(x,t) = –u

x,t+T 

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ThenHis invariant under shift: Leth∈Handτ be a real number. Ifv(x,t) =u(x,t+τ), thenv∈H.His invariant underg: Letu∈Hsuch thatg(u)∈L_{(). Then}_g(u)_∈_{H. Let}

˜

u(x,t) =u(x,t+T_). Then

u=

j>,k

ujk(–)ksinjxexpik

a bt.

Therefore

u∈H ⇐⇒ uj,k=  for any evenk. (.)

LetAbe a linear operator ofHdeﬁned by

D(A) =D(A)∩H,

Au=Au.

ThenAis self-adjoint inHandH∩N(A) ={}, whereN(A) is the kernel ofA. In fact, let

u∈H∩N(A). Then

u=uj,ksinjxexpik

a bt,

j–k

_a

b =  ⇒ uj,k= .

Letjandkbe such that

j–k

_a

b = .

Sincebis even andais odd,kis even. Using (.), we haveuj,k= , and thereforeH∩N(A) =

{}. The eigenvalues ofAarej– (π_Tk), wherejis odd andkis even andH∩N(A) ={}.

Givenu∈H, we write

u=

j>

uj,ksinjxexpi

πkt

T , jis odd,kis even

with

uj,k=u¯j,–k.

Let

E=

u∈H

j,k

j–a

_k

b

· |uj,k|< +∞ ,

(u,v) =

j,k

j–a

_k

b

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where ( , ) is a scalar product onE. With this scalar product,Eis a Hilbert space with a norm

u= (u,u)_, _u∈_E.

Let

ur=

|u|r

 r

, r≥.

By the classical theorem of Riesz (cf.[, p.]), we have

ur≤

πT

 

r

j,k

|uj,k|r

 r

, r≥, r+

 r= .

Since for every>o

jodd,keven



|j_–a_k

b |+

<∞,

it follows that for everyr∈[, +∞), there iscr∈Rsuch that

ur≤cru. (.)

Let

E+=

uu∈E,uj,k=  ifj–

ak b <  ,

E–=

uu∈E,uj,k=  ifj–

a_k

b >  .

Then E=E+⊕E–. Let P+ be the orthogonal projection fromE ontoE+ andP– be the

orthogonal projection fromEontoE–. We can writeu=P+u+P–uforu∈E.

3 Critical point theorem and the proof of Theorem 1.1 Now, we are going to seek a functionuinEsuch that

Au=g(x,t,u). (.)

We consider the associated functional of (.),

I(u) =  

–|ut|+|ux|

dx dt–

G(x,t,u)dx dt

= 

P+u–P–u

–

G(x,t,u)dx dt, (.)

where

G(x,t,u) = u



g(x,t,s)ds.

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We recall the critical point theorem for the indeﬁnite functional (cf.[]). Let

Br=

u∈H|u ≤r, ∂Br=

u∈H|u=r.

Theorem .(Critical point theorem for the indeﬁnite functional) Let E be a real Hilbert space with E=E⊕Eand E=E⊥.Suppose that I∈C(E,R)satisﬁes(PS),and that

(I) I(u) =_(Lu,u) +bu,whereLu=LPu+LPuandLi:Ei→Eiis bounded and

self-adjoint,i= , , (I) bis compact,and

(I) there exist a subspaceE˜⊂Eand setsS⊂E,Q⊂ ˜Eand constantsα>ωsuch that

(i) S⊂EandI|S≥α,

(ii) Qis bounded andI|∂Q≤ω,

(iii) Sand∂Qlink.

Then I possesses a critical value c≥α.

The eigenvalues ofAareλjk =j– (_Tπk), wherejis odd andkis even,j= , , . . . and

k= , , , . . . . ThusA–

 is a compact operator.

It is convenient for the following to rearrange the eigenvaluesλjk, wherejis odd andkis

even, by increasing magnitude: from now on we denote by (η–_i)i≥=λjk<  the sequence

of negative eigenvalues, by (η+_i)i≥=λjk>  the sequence of positive ones, so that

· · · ≤η_i–≤ · · · ≤η–_≤η–_ <η+_ ≤η+_≤ · · · ≤η+_i ≤ · · ·. (.)

We note that each eigenvalue has a ﬁnite multiplicity and thatη–

i →–∞,ηi+→+∞as

i→ ∞.

We shall show that the functionalIsatisﬁes the geometrical assumptions of the critical point theorem for the indeﬁnite functional.

By the following lemma, I(u)∈C_(E,_{R) and the solutions of (.) coincide with the}

nonzero critical points ofI(u).

Lemma . Assume that g satisﬁes(g)-(g).Then I(u)is continuous and Fréchet diﬀer-entiable in E with the Fréchet derivative

I(u)h=

–ut·ht+ux·hx–g(x,t,u)h

dx dt

= (P+u,P+h) – (P–u,P–h) –

g(x,t,u)h dx dt

for all h∈E.Moreover,if we set

F(u) =

G(x,t,u)dx dt,

then F(u)is continuous with respect to weak convergence,F(u)is compact,and

F(u)h=

g(x,t,u)h dx dt for all h∈E.

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For the proof of Lemma ., refer to [].

Lemma . Assume that g satisﬁes(g)-(g).The problem

utt–uxx in E,

u(,t) =u(π,t) = ,

u(x,t+T) =u(x,t),

ut(x, ) =ut(x,T) ∀x∈[,π]

(.)

has only a trivial solution.

Proof LetAu=utt–uxx. SinceH∩N(A) ={},E∩N(A) ={}. Thus (.) has only a trivial

solution.

Lemma . Assume that g satisﬁes(g)-(g).Then I(u)satisﬁes the Palais-Smale condi-tion:If for a sequence(um),I(um)is bounded from above and I(um)→as m→ ∞,then

(um)has a convergent subsequence.

Proof Let (um) be a sequence withI(um)≤MandI(um)→ asm→ ∞. It suﬃces to

show that (um) is bounded. By contradiction, we suppose thatum → ∞asm→ ∞. Let

wm=u_um_m. Thenwm= andwmconverges weakly to an element, sayw. Then we have

M

um

≤ I(um)

um

= 

–(wm)t+(wm)xx

–

G(x,t,um)

um

. (.)

SinceG(x,t,u) is bounded, lettingm∞in (.), we have

≤limP+wm–limP–wm=

–|wt|+|wxx|

. (.)

On the other hand, we have

←− I

_(u

m)

um

=

–(wm)t



+(wm)xx



–

g(x,t,um)

um

. (.)

Lettingm→ ∞in (.), we have

 = lim

m→∞P+wm

_– _lim

m→∞P–wm

₌

–|wt|+|wxx|

=P+w–P–w. (.)

By (.) and (.), limm→∞P+wm =P+w and limm→∞P–wm =P–w. Thus limm→∞wm=w, sowm converges strongly to wand by (.),wis a weak solution

of the problem

wtt–wxx=  inE.

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Let

Q= (B¯R∩E–)⊕ {re|e∈∂B∩E+ <r<R}.

Lemma . Assume that g satisﬁes the conditions(g)-(g).There exist a subspaceE˜⊂E and sets∂Bρ⊂E,Q⊂ ˜E and constantsρ> andα> such that

(i) ∂Bρ⊂E+andI|∂Bρ≥α,

(ii) there aree∈∂B∩E+andR>ρsuch that ifQ= (B¯R∩E–)⊕ {re| <r<R},then

I|∂Q≤,

(iii) there existsu∈E\Qsuch thatu>RandI(u)≤, (iv) ∂Bρand∂Qlink.

Proof (i) Letu∈E+. SinceG(x,t,u) is bounded, there exists a constantC>  such that

I(u) =  P+u

_–

P–u

_–

G(x,t,u)

≥ 

P+u

_–_C

forC> . Then there exist a constantρ>  and a constantα>  such that ifu∈∂Bρ∩E+,

thenI(u)≥α.

(ii) Let us choose an elemente∈B∩E+. Letu∈(B¯r∩E–)⊕ {re| <r}. Thenu=v+w,

v∈Br∩E–,w=re. We note that

ifv∈Br∩E–, then

–|vt|+|vx|

dx dt≤η_i–v_L₍₎< .

Thus we have

I(u) =  r

_–

v

_–

G(x,t,v+re)

≤ 

r

₊

η

–

vL₍₎+C

forC> . Then there existsR>ρ>  such that ifu∈Q= (B¯R∩E–)⊕ {re| <r<R}, then

I(u)|∂Q≤.

(iii) Let us chooseu∈E\Q. By (ii),I(u)≤.

(iv)SandQlink at the set{e}.

Proof of Theorem. By Lemma .,I(u) is continuous andFréchetdiﬀerentiable inE. By Lemma ., the functional I satisﬁes the (PS) condition. We note that I() = . By Lemma ., there are constantsρ> ,α>  and a bounded neighborhoodBρof  such that

I|∂Bρ∩E+≥α, and there aree∈∂B∩E+andR>ρsuch that ifQ= (B¯R∩E–)⊕{re| <r<R}.

Let us set

=γ ∈C(Q,¯ H)|γ = id on∂Q.

By Theorem .,Ipossesses a critical valuec≥α. Moreover,ccan be characterized as

c=inf γ∈maxu∈QI

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Thus we prove thatIhas at least one nontrivial critical point, we denote byu˜ a critical point ofIsuch thatI(u) =˜ c. We claim thatcis bounded. In fact, we have

c≤ max ≤τ≤I(τu),

and by (g),

I(τu) =τ

 P+u

_–

P–u



–

G(x,t,τu)dx dt

≤τu–

G(x,t,τu)dx dt

≤Cτu+C

for some constantC> . Since ≤τ≤,cis bounded:

c<C.˜ (.)

We claim thatu˜is bounded. In fact, by contradiction,u˜tt+u˜xx=g(x,t,u) and, for any˜ K> ,

max|˜u(x,t)|>Kimply that

c=I(u) =˜  

P+u˜–P–u˜

–

G(x,t,u)˜ dx dt

is not bounded, which is absurd to the fact thatc=I(u) is bounded. Thus˜ u˜is bounded, so (.) has at least one bounded weak solution, and hence we prove Theorem ..

Competing interests

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Authors’ contributions

The authors did not provide this information.

Author details

1_{Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea.}2_{Department of Mathematics}

Education, Inha University, Incheon, 402-751, Korea.

Acknowledgements

This work (Choi) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).

Received: 27 June 2013 Accepted: 25 October 2013 Published:25 Nov 2013

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Cite this article as:Jung and Choi:Bounded solutions for the nonlinear wave equation.Boundary Value Problems