## Abstract

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with *x*-dependent coefficients *u*(*x*)*y*_{tt}*−*(*u*(*x*)*y*_{x})_{x}+*au*(*x*)*y*+|*y*|^{p−2} *y*=*f*(*x*, *t*) on (0, *π*)× under the periodic or anti-periodic boundary conditions *y*(0,*t*)=±*y*(*π*, *t*), *y*_{x}(0, *t*)=±*y*_{x}(*π*, *t*) and the time-periodic conditions *y*(*x*, *t*+*T*)=*y*(*x*, *t*), *y*_{t}(*x*, *t*+*T*)=*y*_{t}(*x*, *t*). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For *T*=2*π*/*k*(*k*∈), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with *x*-dependent coefficients.

## 1. Introduction

In this paper, we consider the existence of time-periodic solutions to the following nonlinear wave equation:(1.1)with the periodic or anti-periodic boundary conditions(1.2)and the time-periodic conditions(1.3)Here, *a* is a parameter, *f*(*x*, *t*) is a *T*-periodic (in time) function that is continuous on [0,*π*]×, and *T* is assumed to be commensurable with *π* and to have the form(1.4)

The reason we take *T* in the form (1.4) is only for simplicity. In fact, whenever *k* in (1.4) is rational, there are no small divisors and this allows the use of variational technique, so similar results can also be obtained.

As stated in Barbu & Pavel (1996, 1997*a*,*b*), equation (1.1) describes the forced vibrations of a bounded non-homogeneous string and the propagation of seismic waves in non-isotropic media. More precisely, the vertical displacement *y*(*z*, *t*) at depth *z* and time *t* of a plane seismic wave is described by the equationunder some boundary conditions in *z* and initial conditions in *t*. Here, *ρ* is the rock density and *μ* is the elasticity coefficient. By the change of variable given bywe obtainwhere *u*=(*ρμ*)^{1/2} denotes the acoustic impedance function.

The problem of finding time-periodic solutions to nonlinear wave equations with constant coefficients (i.e. when *u*(*x*)≡1) has attracted much attention since the 1960s (see Cesari 1965; Rabinowitz 1967, 1978; Brézis & Nirenberg 1978; Bahri & Brézis 1980; Brézis 1983; Feireisl 1988*a*; Plotnikov & Yungerman 1988; Craig & Wayne 1993; Berti & Bolle 2003; Favini *et al*. 2005; Berti & Biasco 2006 and the references cited there). Recently, Barbu & Pavel (1996, 1997*a*,*b*) considered the wave equations with *x*-dependent coefficients for the first time. In Barbu & Pavel (1997*b*), the existence and regularity of time-periodic solutions were obtained for the case in which the nonlinear term has sublinear growth and satisfies the global Lipschitz condition. For the case that the nonlinear term has power-law growth, Rudakov (2004) proved the existence of time-periodic solutions. However, all their results are dealing with the Dirichlet boundary-value problem. Not very many results seem to be known for the periodic or anti-periodic boundary-value problem. In Ji & Li (2007), we treated them for the case in which the nonlinear term has sublinear growth and satisfies the global Lipschitz condition. In addition, we also studied the general boundary-value problem in Ji & Li (2006) and Ji (2007), respectively, for sublinear growth and power-law growth nonlinearity. In this paper, we shall establish the existence of time-periodic solutions to wave equation (1.1) under the periodic or anti-periodic boundary conditions (1.2) and the time-periodic conditions (1.3).

Set *Ω*_{T}=(0, *π*)×(0, *T*) and . Throughout this paper, we shall make the following hypotheses:

satisfies 0<

*α*_{0}≤*u*(*x*)≤β_{0}for*x*∈[0,*π*] andThe constant

*p*in the exponent of the power satisfies*p*>2.

On hypothesis (H1), we remark that the condition 0<*α*_{0}≤*u*(*x*)≤*β*_{0} is natural because *u*(*x*) denotes the acoustic impedance function and usually is positive and bounded. However, the assumption ess inf *η*_{u}(*x*)>−*a* is only a technical requirement to avoid the null eigenvalue.

This paper is organized as follows. We first give some preliminaries and the main results in §2. Then we shall study the properties of the wave operator with *x*-dependent coefficients and the existence of weak solutions for the periodic boundary-value problem and anti-periodic boundary-value problem in §3 and §4, respectively. Finally, further discussions and conclusions are given in §5.

## 2. Preliminaries and main results

LetWe definewhere *r*≥1. The space *L*^{r}(*Ω*_{T}) is the closure of *Φ*^{±} in the norm . Let *q* denote the conjugate exponent of *p*, i.e. 1/*p*+1/*q*=1. Then (H2) implies 1<*q*<2. For functions *f*∈*L*^{p}(*Ω*_{T}) and *g*∈*L*^{q}(*Ω*_{T}), we define

The function *y*∈*L*^{p}(*Ω*_{T}) is called a weak solution to problem (1.1)–(1.3) iffor all *φ*∈*Φ*^{±}.

The main goal of this paper is to prove the following existence theorem.

*Assume that T is commensurable with π as in* (*1.4*) *and that hypotheses* (*H1*) *and* (*H2*) *hold*. *Then for all positive numbers d*_{1}, *d*_{2}, *there exists a k*_{0}=*k*_{0}(*d*_{1},*d*_{2})∈ *such that, for any k*≥*k*_{0}, *k*∈ *and any function f*(*x*,*t*) *which is T-periodic in t and continuous on* [0,*π*]× *with* ‖*f*‖_{C}≤*d*_{1}, *the problem* (*1.1*)–(*1.3*) *has a non-trivial weak solution y*∈*L*^{p}(*Ω*_{T}) *such that* .

As in Feireisl (1988*a*), for convenience, in what follows, we shall replace the problem (1.1)–(1.3) by the equivalent 2*π*-periodic problem. To do this, let us define a function Then it is clear that solves the problem (1.1)–(1.3) if and only if *z* is a solution of the problem(2.1)

(2.2)

(2.3)

From now on, we shall write *Ω*, *L*^{p}(*Ω*), ‖.‖_{p} instead of *Ω*_{2π}, *L*^{p}(*Ω*_{2π}), , respectively. Define the linear operator with *x*-dependent coefficients byfor all *φ*∈*Φ*^{±}. We denote and suppose that *A*^{±} is the extension of in *L*^{2}(*Ω*), i.e.

For the study of periodic solutions to (2.1)–(2.3), we need to use the following complete orthonormal system of eigenfunctions {*ψ*_{m}*φ*_{n}; *m*∈,*n*∈^{0}={0}∪} in *L*^{2}(*Ω*) (see Yosida 1980), whereand *λ*_{n}, *φ*_{n} are given by the Sturm–Liouville problem(2.4)

(2.5)

(2.6)

The inner product in *L*^{2}(0, *π*) is defined byThus, .

In order to characterize the asymptotic formulae of eigenvalues *λ*_{n}, we set *z*_{n}(*x*)=(*u*(*x*))^{1/2}*φ*_{n}(*x*), then *z*_{n} satisfies the Sturm–Liouville problem(2.7)

(2.8)

(2.9)

*The eigenvalues of* (*2.7*)–(*2.9*) *satisfy λ*_{n}≥*ρ*+*a*>0 *for all n*∈^{0}. *In particular, for anti-periodic boundary conditions*, *i.e.* ‘−’ *is present in* (*2.8*) *and* (*2.9*), *then λ*_{n}≥*ρ*+*a for all n*∈^{0}.

The proof is similar to that of lemma 2.3 in Ji & Li (2007), and here it is omitted for brevity. ▪

The further investigation shows that the eigenvalues *λ*_{n} of (2.4)–(2.6) or (2.7)–(2.9) are provided with different form for the periodic boundary conditions and anti-periodic boundary conditions. Therefore, in what follows, we shall study the properties of the wave operator and the existence of weak solutions to (2.1)–(2.3) based on the type of boundary conditions, respectively.

## 3. Periodic boundary-value problem

Consider the periodic boundary-value problem(3.1)(3.2)(3.3)In this case, the Sturm–Liouville problem (2.7)–(2.9) can be rewritten as(3.4)

(3.5)

(3.6)

It is known (see Marchenko 1977; Dymarskii 2002) that the eigenvalues of problem (3.4)–(3.6) are arranged as an increasing unbounded sequence so that (i) if the equality sign is present then the corresponding eigenvalue is double and (ii) the zeros of an eigenfunction on the segment [0,*π*) are equal to 2*n*, where *n* is the number of the corresponding eigenvalue.

### (a) Asymptotic formulae of eigenvalues

*Let* *and* *denote the eigenvalues and the corresponding real orthonormal eigenfunctions of* (*3.4*)–(*3.6*), *respectively. Then*, *for all n*∈, *we have*

First, we prove the lower bound. By lemma 2.3, we know that holds for all *n*∈. In (3.4), we introduce the Prüfer transformationwith *r*(*x*)>0. It is easy to obtain that(3.7)Note that has exactly 2*n* zeros in [0,*π*) and we denote these by *τ*_{1}<*τ*_{2}<⋯<*τ*_{2n}. Correspondingly, we denote *θ*_{i}=*θ*(*τ*_{i}) for *i*=1,…,2*n*. Thus, we may take *θ*_{i}=*iπ* for *i*=1, 2,…,2*n*. If *τ*_{1}=0, (3.5) implies *θ*(*π*)=(2*n*+1)*π*. Thus, we have *θ*(*π*)−*θ*(0)=2*nπ*. On the other hand, if *τ*_{1}>0, we denote *θ*(0)=*θ*_{0}∈(0,*π*), (3.5) implies *θ*(*π*)=2*nπ*+*θ*_{0}. Thus, we also have *θ*(*π*)−*θ*(0)=2*nπ*. Integration of (3.7) over [0,*π*] yields thati.e.

In what follows, we prove the upper bound. By using the Prüfer transformationwith *r*(*x*)>0 in (3.4), we have(3.8)

As above, we can obtain *θ*(*π*)−*θ*(0)=2*nπ*. Integration of (3.8) over [0,*π*] yields thatwhich is equivalent towhere and .

Solving for by using the quadratic formula yields that(3.9)

The elementary inequality gives(3.10)

Squaring (3.9) and using (3.10), we obtain(3.11)i.e.which complete the proof. ▪

Let . By lemma 3.1, we can obtainwhich holds for all *n*∈. On the other hand, let *ρ*_{0}=*ρ*+*a*, then we havewhich holds for all *n*∈. Hence, we have the following result.

*Let u satisfy* (*H1*). *Then the eigenvalues* (*where λ*_{0} *is denoted by* *for convenience*) *of* (*3.4*)–(*3.6*) *have the form*(3.12)*where*

### (b) Properties of wave operator

First, similar to the proof in Ji & Li (2007), we can establish the following properties of wave operator *A*^{+} by using the asymptotic formulae of .

*Assume that T is commensurable with π as in* (*1.4*) *and u satisfies* (*H1*). *Then we have*

*R*(*A*^{+})*is closed in L*^{2}(*Ω*),*A*^{+}*is self-adjoint and*(*A*^{+})^{−1}∈*L*(*R*(*A*^{+}),*R*(*A*^{+})),*the null space**is finite dimensional*,*and**and*.

Furthermore, in order to prove our main result, we also need some other properties for wave operator *A*^{+} which will be established as follows.

Let and , where , . Then *L*^{2}(*Ω*) can be expressed asHere we remark that the subspace *N*_{2} is the close-to-null space of the wave operator *A*^{+}, that is to say, *N*_{2} would be the null space in the constant coefficient case. However, in this case, there are no small divisors on *N*_{2}. This is because, on *N*_{2}, the eigenvalues of *A*^{+} are equal to , which is bounded by .

*There exists a natural number k*_{1} *such that for all natural numbers k*≥*k*_{1} *the following estimate holds:*(3.13)*where* 1<*r*≤2 *and C*_{0}=*C*_{0}(*k*_{1}) *is a constant independent of v and k*.

Let *v*∈*N*_{3}, be the Fourier coefficients of *v* and . Noting that the eigenvalues of operator *A*^{+} are , *m*∈,*n*∈^{0}, by the Hölder and Hausdorff–Young inequalities, we have(3.14)Now, we need to prove the convergence of the following series:In view of lemma 3.2, we haveDenote , then(3.15)holds for (*m*,*n*)∈*N* and *m*≠0. This implies thatexists. In addition, since , it is easy to know that *δ*(*k*)≠0. It follows from (3.15) that there exists a natural number *k*_{1} such that for all natural numbers *k*≥*k*_{1} and *m*≠0 the following estimate holds:Thus, *δ*(*k*)≥1/2 for all *k*≥*k*_{1} andThis implies the series *I* is convergent. Taking into account (3.14), the estimate (3.13) is obtained. Thus, the proof is completed. ▪

*The operator* (*A*^{+})^{−1}: *N*_{3}→*N*_{3} *is totally continuous*.

In the proof of lemma 3.4, we established the convergence of the seriesSetting *r*=2, we obtain the assertion of the lemma. ▪

*The eigenvalues of* *have finite multiplicity*.

Consider the equationwhich is equivalent toIt follows from (3.12) that the right-hand side in this relation is bounded and can take only a finite integral number. Now, note that for *l*∈\{0} the equation(3.16)has at most a finite number of solutions (*m*,*n*)∈×^{0}, which is obtained by an exhaustive search of all integer divisors of *l*. On the other hand, since 2*n*≠*k*|*m*|, it is obvious that equation (3.16) has no solutions for *l*=0. The proof is completed. ▪

### (c) Existence of weak solutions

Let the finite dimensional space *E*_{n}=*N*(*A*^{+})⊕*N*_{2n}⊕*N*_{3n}, where and . We define the energy functional *F*: *E*_{n}→ byIt is easy to see that

In what follows, we shall prove theorem 2.2 for the periodic boundary-value problem (3.1)–(3.3). The proof is divided into two steps.

*Step* 1. Existence of a critical point of *F* on *E*_{n}. The proof is based on the following lemma.

*Suppose that a finite-dimensional space E is expressed as the direct sum of pairwise-orthogonal subspaces, E*=*V*_{1}⊕*V*_{2}⊕*V*_{3}. *Suppose that S is a sphere in E centred at zero. Suppose that the functional F: E*→ *satisfies the following conditions*:

*and*.

*Then there exists a critical point v*_{0}∈*E of the functional F such that F*(*v*_{0})∈[*c*,*d*].

The proof can be found in the appendix of Feireisl (1988*b*). Here, we omit it. ▪

For any *c*∈, by *G*_{c} and *L*_{c} we denote the subspace of *E*_{n} that are the linear hulls of the eigenfunctions of the operator *A*^{+} whose eigenvalues are not less and less, respectively, than the number *c*.

For any *c*∈ and *v*∈*G*_{c}, we havewhere the constants *C*_{4}, *C*_{5} are independent of *n*, *c* and *k*. We denoteThen(3.17)The number *m*(*c*,*d*_{1}) is independent of *n* and *k*.

We take *v*∈*N*_{3n}. Suppose that are the Fourier coefficients of *v*, i.e.(3.18)We denoteFor *k*≥*k*_{1}, the estimate (3.13) and the interpolational inequality yield the estimate (see Feireisl 1988*a* or Rudakov 2004)(3.19)where the constant *C*_{6} is independent of *n*, *v* and *k* and *β*=*s*(*p*−2)/*p*. If *s*∈(1,*p*/(*p*−2)), then(3.20)

*For any c*∈, *there exists k*_{0}=*k*_{0}(*c*,*d*_{1})∈ *such that if k*≥*k*_{0} *and k*∈ *then for any* *the following inequality holds:*

Let *v*∈*L*_{0} and . We denote the Fourier coefficients of *v* that are calculated by formulae (3.18). Noting that *L*_{0}⊂*N*_{3n}, we have(3.21)where *λ* denotes the greatest negative eigenvalue of *A*^{+}. Noting thatthis combining with (3.12) yields(3.22)Taking into account (3.20), (3.21) and (3.22) imply the assertion of the lemma. ▪

Let *E*=*E*_{n}, *V*_{1}=*L*_{λ}, *V*_{3}=*G*_{0} and *V*_{2} denote the linear hull of the eigenfunctions of operator *A*^{+} with eigenvalue *λ*. It follows from lemma 3.6 that dim *V*_{2}<∞. In what follows, we shall verify the validity of the assumptions in lemma 3.7.

We take an arbitrary number *c*<*m*(0,*d*_{1}) and denote *γ*(*c*,*d*_{1})=min(*c*,*m*(*λ*,*d*_{1}))−1. It follows from (3.17) thatLet . Noting that *L*_{0}=*V*_{1}⊕*V*_{2}, by lemma 3.8, we conclude that there exists a *k*_{0}=*k*_{0}(*c*,*d*_{1}) such that if *k*≥*k*_{0} and *k*∈ thenIn addition, from (3.17) we can also obtain

Conditions (1) and (2) of lemma 3.7 are easy to be verified. Thus, all the assumptions of lemma 3.7 are satisfied. Hence, it follows that there exists a critical point *v*_{n}∈*E*_{n} of *F* on *E*_{n} such that *F*(*v*_{n})∈[γ(*c*,*d*_{1}),*c*]. Therefore, we obtain(3.23)

(3.24)

*Step* 2. Passage to the limit as . Replacing *ω* in (3.23) by *v*_{n} and combining with (3.24), we obtainwhich implies(3.25)(3.26)where *C*_{9}, *C*_{10}, *C*_{11} are positive constants independent of *n*.

The estimate (3.25) shows that there exists a subsequence that is denoted by itself for simplicity such that *v*_{n}→*v* weakly in *L*^{p}(*Ω*). Furthermore, note thatwhich combining with (3.25) shows that |*v*_{n}|^{p−2}*v*_{n} is bounded in *L*^{q}(*Ω*). Therefore, there exists a subsequence that is denoted by itself for simplicity such that |*v*_{n}|^{p−2}*v*_{n}→*h* weakly in *L*^{q}(*Ω*).

In what follows, we prove that *v* is a weak solution of problem (3.1)–(3.3). To do this, we rewritewhere *v*_{1},*v*_{1n}∈*N*_{1}, *v*_{2},*v*_{2n}∈*N*_{2}, *v*_{3},*v*_{3n}∈*N*_{3}.

It is readily seen that *v*_{kn}→*v*_{k} weakly in *L*^{2}(*Ω*). For fixed *ω*∈*E*_{n}, we pass to the limit in (3.23) and obtain(3.27)

In what follows, we prove that *h*=|*v*|^{p−2}*v* using the monotonicity method (see Feireisl 1988*a*). Suppose that and are the Fourier coefficients of *v*_{n} and *v*, respectively. As in Feireisl (1988*a*), we substitutewhere , into (3.23). By (3.25) and (3.19), we have(3.28)On the other hand, we know that(3.29)LetThe combination of (3.28) and (3.29) yieldswhich implieswhere *C*_{13} is a constant independent of *n*. It follows from the last inequality that(3.30)

Then, setting *ω*=*v*_{n} in (3.23) and using (3.30), we obtainSubstituting *ω*=*v*_{n} into (3.27) and letting *n*→∞, we obtainThus, we obtain(3.31)Note that the eigenvalues of *A*^{+} on *N*_{2} are positive and function *y*=|*x*|^{p−2}*x* is strictly increasing. Therefore, for any *w*∈*L*^{p}(*Ω*), we haveBy passing to the limit as *n*→∞, we obtain

Taking *w*=*v*+*λϕ*, letting *λ*→0, and using the standard arguments, we obtain *h*=|*v*|^{p−2}*v*. It follows from (3.27) that *v* is a weak solution of problem (3.1)–(3.3). The inequality ‖*v*‖_{p}≥*d*_{2} follows from (3.26) and (3.31) for a sufficiently large |*c*|.

## 4. Anti-periodic boundary-value problem

Consider the following anti-periodic boundary-value problem:(4.1)(4.2)(4.3)In this case, the Sturm–Liouville problem (2.7)–(2.9) can be rewritten as(4.4)

(4.5)

(4.6)

It is known (see Marchenko 1977; Dymarskii 2002) that the eigenvalues of problem (4.4)–(4.6) are arranged as an increasing unbounded sequence so that (i) if the equality sign is present then the corresponding eigenvalue is double and (ii) the zeros of an eigenfunction on the segment [0,*π*) are equal to 2*n*+1, where *n* is the number of the corresponding eigenvalue.

### (a) Asymptotic formulae of eigenvalues

*Let* *and* , *denote the eigenvalues and the corresponding real orthonormal eigenfunctions of* (*4.4*)–(*4.6*), *respectively. Then, for all n*∈^{0}, *we have*

First, we prove the lower bound. By lemma 2.3, we know that holds for all *n*∈^{0}. Similar to the proof of the lower bound in lemma 3.1, introducing the Prüfer transformationwith *r*(*x*)>0 in (4.4), we have(4.7)Since has exactly 2*n*+1 zeros in [0,*π*), we denote these zeros by *τ*_{1}<*τ*_{2}<⋯<*τ*_{2n}<*τ*_{2n+1}. Correspondingly, we denote *θ*_{i}=*θ*(*τ*_{i}). Thus, we may take *θ*_{i}=*iπ* for *i*=1, 2,…,2*n*+1. If *τ*_{1}=0, (4.5) implies *θ*(*π*)=(2*n*+2)*π*. Thus, we have *θ*(*π*)−*θ*(0)=(2*n*+1)*π*. On the other hand, if *τ*_{1}>0, we denote *θ*(0)=*θ*_{0}∈(0,*π*), (4.5) implies *θ*(*π*)=(2*n*+1)*π+θ*_{0}. Thus, we also have *θ*(*π*)−*θ*(0)=(2*n*+1)*π*. Integration of (4.7) over [0,*π*] yields thati.e.

In what follows, we prove the upper bound. By using the Prüfer transformationwith *r*(*x*)>0 in (4.4), we have(4.8)As above, we can obtain *θ*(*π*)−*θ*(0)=(2*n*+1)*π*. Integration of (4.8) over [0,*π*] yields thatwhich is equivalent towhere and . Then by (3.11), we obtainThe proof is completed. ▪

Thus by lemma 4.1, let *ρ*_{0}=*ρ*+*a*, we haveOn the other hand, let . Then we haveHence, we have the following result.

*Let u satisfy* (*H1*). *Then the eigenvalues* *of* (*4.4*)–(*4.6*) *have the form**where*

### (b) Properties of wave operator

In order to prove our main result, we need to use the following properties of wave operator *A*^{−} that can be proved along the lines given in Ji & Li (2007).

*Assume that T is commensurable with π as in* (*1.4*) *and u satisfies* (*H1*). *Then we have*

*R*(*A*^{−})*is closed in L*^{2}(*Ω*),*A*^{−}*is self-adjoint and*(*A*^{−})^{−1}∈*L*(*R*(*A*^{−}),*R*(*A*^{−})),*the null space**is finite dimensional*,*and*,

*and*.

In addition, along the line of §3, we can also obtain some other properties of wave operator *A*^{−}, which are enumerated in what follows.

Let and , where , . Then *L*^{2}(*Ω*)=*N*_{1}⊕*N*_{2}⊕*N*_{3}, and we have the following lemmas.

*There exists a natural number k*_{1} *such that for all natural numbers k*≥*k*_{1} *the following estimate holds:**where* 1<*r*≤2 *and C*_{0}=*C*_{0}(*k*_{1}) *is a constant independent of v and k*.

*The operator* (*A*^{−})^{−1}:*N*_{3}→*N*_{3} *is totally continuous*.

*The eigenvalues of* *have finite multiplicity*.

### (c) Existence of weak solutions

By using the properties of *A*^{−}, along the lines of §3, it is not difficult to prove that theorem 2.2 holds for anti-periodic boundary-value problem (4.1)–(4.3). Here, we omit the details.

## 5. Further discussion and conclusion

We have established the existence of time-periodic solutions of problem (1.1)–(1.3) by using the properties of wave operator *A*^{±}. Furthermore, on the regularity, it is also easy to prove that *y*∈*L*^{∞}(*Ω*) (see Feireisl 1988*a*), by taking into consideration the monotonicity of nonlinearity and the finiteness of the dimension of null space *N*(*A*^{±}). In addition, note that our proof is not dependent on the homogeneity of the nonlinear term, therefore our result can be extended to the following more general problem:(5.1)(5.2)(5.3)where *h*(*x*, *t*, *y*) is assumed to satisfy the following conditions.

The continuity condition:

*h*(*x*,*t*,*y*) is continuous on the set [0,*π*]××.The periodicity condition:

*h*(*x*,*t*+*T*,*y*)=*h*(*x*,*t*,*y*) for all (*x*,*t*,*y*)∈[0,*π*]××, where .The monotonicity condition:

*h*(*x*,*t*,*y*) is non-decreasing in*y*for all (*x*,*t*)∈[0,*π*]×.The growth condition: there exist positive constants

*A*_{1},*A*_{2},*A*_{3},*A*_{4},*δ*such thatholds for all (*x*,*t*,*y*)∈[0,*π*]××.

*Assume that T is commensurable with π as in* (*1.4*) *and that* (*H1*) *and* (*H2*) *hold*. *Then for any d*>0, *there exists a k*_{0}=*k*_{0}(*d*,*A*_{1},*A*_{2},*A*_{3},*A*_{4},*δ*)∈ *such that, for any k*≥*k*_{0}, *k*∈ *and any function h*(*x*,*t*,*y*) *which satisfies* (*C1*)–(*C4*), *the problem* (*5.1*)–(*5.3*) *has a weak solution y*∈*L*^{p}(*Ω*_{T}) *such that* .

Along the line of the proof of theorem 2.2, it is easy to prove theorem 5.1. Here, we omit the details.

In particular, we consider the following problem:(5.4)(5.5)(5.6)where *h*_{1}(*x*, *t*, *y*) is assumed to satisfy the following conditions.

The function

*h*_{1}is periodic (in time) with period*T*, continuous in [0,*π*]××, non-increasing in*y*for (*x*,*t*)∈[0,*π*]× and satisfiesfor some positive constants*B*_{1},*B*_{2}and*γ*<*p*−1.

Denote *h*_{2}(*x*, *t*, *y*)=|*y*|^{p−2} *y*−*h*_{1}(*x*, *t*, *y*). It is obvious that *h*_{2}(*x*, *t*, *y*) satisfies conditions (C1)–(C3). LetSince *γ*<*p*−1, then there exists a *K*>0 such thatThus, we have(5.7)On the other hand, we can obtain(5.8)In addition, we havei.e.(5.9)The inequalities (5.7)–(5.9) show that the function *h*_{2}(*x*, *t*, *y*) satisfies condition (C4). Therefore, we have the following result.

*Assume that T is commensurable with π as in* (*1.4*) *and that hypotheses* (*H1*) and (*H2*) *hold*. *Then for any d*>0, *there exists a k*_{0}=*k*_{0}(*d*,*B*_{1},*B*_{2},*γ*)∈ *such that, for any k*≥*k*_{0}, *k*∈ *and any function h*_{1}(*x*,*t*,*y*) *which satisfies* (*C5*), *the problem* (*5.4*)–(*5.6*) *has a weak solution y*∈*L*^{p}(*Ω*_{T}) *such that* .

## Acknowledgments

This work was supported by NSFC grant 10801060, SRFDP grant 20070183052, the 985 Project of Jilin University and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University. The author sincerely thanks Professor Yong Li for his instructions and many useful suggestions. The author also wishes to thank the anonymous referees for their careful reading of the manuscript and for their helpful suggestions and comments which improved this paper.

## Footnotes

- Received June 30, 2008.
- Accepted November 4, 2008.

- © 2008 The Royal Society