Periodic Solutions to Klein–Gordon Systems with Linear Couplings

Jianyi Chen; Zhitao Zhang; Guijuan Chang; Jing Zhao

doi:10.1515/ans-2021-2138

Article Open Access

Periodic Solutions to Klein–Gordon Systems with Linear Couplings

Jianyi Chen , Zhitao Zhang , Guijuan Chang and Jing Zhao

Published/Copyright: July 17, 2021

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From the journal Advanced Nonlinear Studies Volume 21 Issue 3

Abstract

In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories

{ u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 ,

where u,v satisfy the Dirichlet boundary conditions on spatial interval [0,π], b>0 and f,g are 2⁢π-periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions (uε,vε) with time period 2⁢π for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε→0, (uε,vε) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

Keywords: Wave Equation; Variational Method; Klein–Gordon System; Periodic Solutions

MSC 2010: 35B10; 35L51; 58E30

1 Introduction

In this paper, we consider the important nonlinear Klein–Gordon system(1.1)

(1.1a){ut⁢t-ux⁢x+b⁢u+ε⁢v+f⁢(t,x,u)=0,t∈R,x∈[0,π],vt⁢t-vx⁢x+b⁢v+ε⁢u+g⁢(t,x,v)=0,t∈R,x∈[0,π],

satisfying the Dirichlet boundary value conditions on the 𝑥-axis

(1.1b)u⁢(t,0)=u⁢(t,π)=0,v⁢(t,0)=v⁢(t,π)=0,t∈R,

and periodic conditions with respect to the time variable 𝑡,

(1.1c)u⁢(t+2⁢π,x)=u⁢(t,x),v⁢(t+2⁢π,x)=v⁢(t,x),t∈R,x∈[0,π],

where u⁢(t,x) and v⁢(t,x) are the relativistic wave functions generated by the interaction of two mass fields, b>0 and b stands for the mass; 𝜀 denotes the strength of the fields coupling, and 𝜀 is assumed to be sufficiently small. The nonlinear forced terms f⁢(t,x,u), g⁢(t,x,v) are 2⁢π-periodic in 𝑡. We study the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem (1.1) as 𝜀 goes to 0.

The coupled Klein–Gordon system in the general form

(KG)ut⁢t-ux⁢x=Hu⁢(u,v),vt⁢t-vx⁢x=Hv⁢(u,v)

is deeply connected with many branches of mathematical physics, such as relativistic physics and quantum field theories. For instance, with the proper choice of the potential function H⁢(u,v), system (KG) was used to describe the long-wave dynamics of two coupled one-dimensional periodic chains in the bi-layer materials or the spinless relativistic composite particles (see [1, 18]). Moreover, variations of such systems were also proposed in the work of Klainerman and Tataru [20] as important models to investigate the Yang–Mills equations under the Coulomb gauge condition. The solvability of (KG) depends upon the nature of the nonlinearities and the type of the boundary conditions. Many interesting theoretical and numerical results can be found in [2, 15, 19, 27, 33, 34, 39] and the monograph of Shatah–Struwe [32] which contains more extensive references.

It is an important work to study the existence and regularity of time-periodic solutions for the Dirichlet problem of (KG) with the gradient of the potential function H⁢(u,v) having the interesting coupling form

∇⁡H⁢(u,v)=(-b⁢u-ε⁢v-f⁢(t,x,u),-b⁢v-ε⁢u-g⁢(t,x,v)).

When ε=0, equations (1.1a) are two copies of nonlinear wave equations

(W1)ut⁢t-ux⁢x+b⁢u+f⁢(t,x,u)=0, ⁢t∈R, 0<x<π,

(W2)vt⁢t-vx⁢x+b⁢v+g⁢(t,x,v)=0, ⁢t∈R, 0<x<π.

It is well known that even the existence of time-periodic solutions for single wave equation is difficult to study. Since the seminal work [28] of P. H. Rabinowitz, several tools in nonlinear analysis are developed by H. Brézis, L. Nirenberg, J. M. Coron, K. C. Chang, J. Mawhin, M. Schechter, S. J. Li et al. to obtain the existence and multiplicity results of the periodic solutions for the scalar wave equations with various types of nonlinearities. We refer to [5, 6, 7, 8, 14, 16, 17, 21, 25, 28, 29, 35, 36] for the one-dimensional problem and [9, 10, 11, 26, 30, 31] for higher-dimensional cases. The existence of solutions with time period 𝑇 to such kinds of wave equations depends upon the nature of the parameter 𝑏, period 𝑇 and the nonlinearities. All the above results require the crucial condition that Tπ is rational, and we sometimes take T=2⁢π for simplicity. When 𝑇 is an irrational multiple of 𝜋, we are led to the problems of small divisors which are difficult to deal with (see [3, 4, 37] for examples).

As ε≠0, the solvability of system (1.1) is more complicated because of the presence of the linear coupling terms. We need a more delicate analysis to study the behavior of the interaction between the two linear coupling terms. Berkovits and Mustonen [2] used the topological degree theory and continuation principle to obtain at least one weak solution (u,v) with time period 2⁢π for the system

(KGλ⁢μ){ut⁢t-ux⁢x+λ⁢v+f⁢(t,x,u)=h1⁢(t,x),t∈R,x∈[0,π],vt⁢t-vx⁢x+μ⁢u+g⁢(t,x,v)=h2⁢(t,x),t∈R,x∈[0,π],

where λ⁢μ<0, h1,h2∈L2⁢([0,2⁢π]×[0,π]) and f,g satisfy some linear growth conditions. The assumption λ⁢μ<0 required in [2] plays a crucial role in calculating the degree and getting a priori bounds of solutions for the corresponding homotopy equations.

Recently, Yan, Ji and Sun [40] used the change of degree argument to prove the existence of time-periodic weak solutions for some coupled Klein–Gordon systems with variable coefficients when the forced terms satisfy some sublinear conditions.

To the best of our knowledge, little further progress has been made on the study of the existence and regularity of periodic solutions for (KGλ⁢μ) with superlinear forced terms. In the present work, we study such a superlinear problem and consider the situation λ⁢μ>0 by variational methods. We focus on the case of λ=μ because the variational structure is required. Our results are in three aspects:

existence of the time-periodic weak solutions for (1.1),
asymptotic behavior of the weak solutions as ε→0,
higher regularity of the solutions.

Let Ω=[0,2⁢π]×[0,π]. We say that (u,v)∈L2⁢(Ω)×L2⁢(Ω) is a weak solution to system (1.1) provided that

∬Ω[u⁢(φt⁢t-φx⁢x)+b⁢u⁢φ+ε⁢v⁢φ+f⁢(t,x,u)⁢φ]⁢dt⁢dx=0,

∬Ω[v⁢(ψt⁢t-ψx⁢x)+b⁢v⁢ψ+ε⁢u⁢ψ+g⁢(t,x,v)⁢ψ]⁢dt⁢dx=0

for all functions 𝜑 and 𝜓 satisfying conditions (1.1b), (1.1c) and belonging to the space 𝐻 which is defined by (2.1).

Let σ⁢(L) be the set of eigenvalues of the d’Alembert operator L=∂t2-∂x2 subject to conditions (1.1b) and (1.1c), and denote by ker⁡L the kernel of the operator 𝐿. It is well known that σ⁢(L) consists of the isolated numbers λj⁢k=j2-k2 for j∈Z+ and k∈Z. We see:

0∈σ⁢(L), and the multiplicity of eigenvalue λj0⁢k0=0 is infinite since there exists an infinite number of j0 and k0 such that j02-k02=0; that is, ker⁡L is an infinite-dimensional space;
all the nonzero eigenvalues of 𝐿 are of finite multiplicity, and they tend to +∞ or -∞;
for any number 𝑏 satisfying -b∉σ⁢(L), there exists a constant η>0 such that
(1.2)|λj⁢k+b|≥η for all⁢j∈Z+,k∈Z,
noting that the eigenvalues λj⁢k of the operator 𝐿 are isolated.

1.1 Existence and the Asymptotic Behavior of the Solutions for (1.1)

The first result of this paper is the following theorem concerning the existence of the time-periodic weak solutions for (1.1).

Theorem 1.1

Let b>0 and -b∉σ⁢(L), and f,g∈C⁢(Ω×R,R) are assumed to be 2⁢π-periodic in 𝑡 and satisfy the following superlinear growth and monotonicity conditions:

there exist p>1, q>1 and c0>0 such that, for all t,x,ξ,
|f⁢(t,x,ξ)|≤c0⁢(1+|ξ|p) and |g⁢(t,x,ξ)|≤c0⁢(1+|ξ|q);
f⁢(t,x,ξ)=o⁢(|ξ|) and g⁢(t,x,ξ)=o⁢(|ξ|) as ξ→0 uniformly in (t,x);
(Ambrosetti–Rabinowitz condition)
(p+1)⁢F⁢(t,x,ξ)≤f⁢(t,x,ξ)⁢ξ and (q+1)⁢G⁢(t,x,ξ)≤g⁢(t,x,ξ)⁢ξ
for all 𝑡, 𝑥 and 𝜉, where F⁢(t,x,ξ)=∫0ξf⁢(t,x,s)⁢ds and G⁢(t,x,ξ)=∫0ξg⁢(t,x,s)⁢ds;
(monotonicity) f⁢(t,x,ξ) and g⁢(t,x,ξ) are nondecreasing in 𝜉.

Then there exists ε0>0 such that, for |ε|<ε0, system (1.1) has at least one nontrivial weak solution

(u,v)∈L2⁢(Ω)×L2⁢(Ω)

with time period 2⁢π.

From 2, we infer f⁢(t,x,0)=0 and g⁢(t,x,0)=0, which implies that (u,v)=(0,0) is a solution for system (1.1). We should point out that if (u,v) satisfies (1.1a) and u≢0, then we also have v≢0 due to the structure of system (1.1a). In other words, problem (1.1) possesses no semi-trivial solution of type (u,0) or (0,v).

Remark 1.2

By virtue of 1–4, an explicit computation shows some useful facts for proving Theorem 1.1.

F⁢(t,x,ξ)≥0 and G⁢(t,x,ξ)≥0 for all (t,x,ξ).
There are positive numbers c1 and c2 such that
(1.3)F⁢(t,x,ξ)≥c1⁢|ξ|p+1-c2,G⁢(t,x,ξ)≥c1⁢|ξ|q+1-c2,
and there are constants r¯,c3>0 such that
F⁢(t,x,ξ)≥c3⁢|ξ|p+1,G⁢(t,x,ξ)≥c3⁢|ξ|q+1 for⁢|ξ|≥r¯;
furthermore, it follows from 3 and 4 that
limξ→+∞⁡f⁢(t,x,ξ)=limξ→+∞⁡g⁢(t,x,ξ)=+∞.
We have
F⁢(t,x,ξ)/ξ2→0 and G⁢(t,x,ξ)/ξ2→0 uniformly in⁢(t,x)⁢as⁢ξ→0;
moreover, for each ν>0, there exists a positive number Cν such that
(1.4)|F⁢(t,x,ξ)|≤ν⁢ξ2+Cν⁢|ξ|p+1 and |G⁢(t,x,ξ)|≤ν⁢ξ2+Cν⁢|ξ|q+1.

It is well known that such assumptions as 1–3 are also of great use in solving the nonlinear elliptic equations and the linearly coupled elliptic systems

{-Δ⁢u+u=f⁢(u)+λ⁢v,x∈RN,-Δ⁢v+v=g⁢(v)+λ⁢u,x∈RN

(see [13, 12, 38] and the references therein). We should point out that, in contrast to the elliptic equations and systems, we may face the following difficulties in the problem of finding periodic solutions for the Klein–Gordon system (1.1).

The d’Alembert operator L=∂t2-∂x2 possesses infinitely many eigenvalues going from -∞ to +∞ so that the positive part and the negative part of the spectrum of 𝐿 are all infinite-dimensional spaces, and the functional Φ corresponding to (1.1) stated in Section 2 is neither bounded from above nor from below. Furthermore, because the kernel of 𝐿 is infinite-dimensional, the operator 𝐿 and its inverse are not compact. This fact gives rise to considerable difficulties in solving the strong indefinite problem (1.1). Because of the lack of compactness properties, the embedding estimates and methods used in [13, 12, 38] are invalid here.
Due to the linear coupling effects, it is hard to obtain the energy estimate and convexity properties of the corresponding functional Φ for system (1.1). Many troubles stem from the coupling interplay between the two scalar functions 𝑢 and 𝑣. The main challenges in constructing the time-periodic solutions for (1.1) are to control the energy of Φ in some proper working spaces, and to estimate the asymptotic behavior of the components in the kernel of 𝐿.

Since the solutions for (1.1) obtained in Theorem 1.1 are dependent of 𝜀, we are interested in considering the asymptotic behavior of these solutions as ε→0.

Theorem 1.3

Under the conditions of Theorem 1.1, assume (uε,vε) is a solution of (1.1) obtained in Theorem 1.1 for |ε|<ε0. Let εn∈(-ε0,ε0) be any sequence with εn→0 as n→∞. Then, passing to a subsequence, (uεn,vεn) converge strongly to (U0,V0) in L2⁢(Ω)×L2⁢(Ω) as n→∞, where U0 is a weak solution of (W1), and V0 is a weak solution of (W2) respectively.

1.2 Regularity Results

Another problem that we study is the higher regularity of the solutions to system (1.1). We obtain the L∞ bound of the periodic solutions for (1.1) based on some precise descriptions of the energy estimates.

Theorem 1.4

Suppose that the conditions of Theorem 1.1 are satisfied. Then there exists ε0>0 such that, for |ε|<ε0, the solution of (1.1) lies in L∞⁢(Ω)×L∞⁢(Ω), where L∞⁢(Ω) is the Lebesgue space equipped with the norm

∥w∥L∞=inf⁡{C≥0:|w⁢(t,x)|≤C⁢for almost every⁢(t,x)∈Ω}<∞.

Under some more restrictive assumptions on 𝑓 and 𝑔, we have the following theorem.

Theorem 1.5

Under the conditions of Theorem 1.1, we assume in addition that f,g∈C1⁢(Ω×R,R), and f⁢(t,x,ξ), g⁢(t,x,ξ) are strictly increasing in 𝜉 for all (t,x)∈Ω. Then, for sufficiently small 𝜀, the solution of (1.1) is continuous on Ω.

Our idea for this theorem is motivated by the works of Rabinowitz [28, 29] and Brézis–Nirenberg [6] which are devoted to the regularity of the solutions for scalar wave equations wt⁢t-wx⁢x=h⁢(t,x,w). The above authors decompose 𝑤 into a regular term and a null term via the representation theorem (see [29, Lemma 2.13]). The regular term can be formulated as an integral expression which is continuous in 𝑡 and 𝑥. On the other hand, the null term can be controlled by some delicate integral and pointwise estimates, and then the C0-regularity of the solution is guaranteed by a continuity argument.

However, due to the presence of the linear coupling terms, the previous estimates developed in [6, 28, 29] cannot be applied directly to study the higher regularity of the solutions for system (1.1). Especially, the interplay among the linear coupling null terms is hard to control. To get around this difficulty, a careful calculation and some more precise energy estimates such as (6.3) and (6.4) are required to analyze the interaction between the linear coupling null terms, provided that 𝜀 is sufficiently small.

We conclude this section by illustrating our strategies to tackle the above problems and listing the sketch of the proof of the main results.

First, we prove the existence of the weak solutions for (1.1) by constructing the critical points of the functional Φ defined by (2.2) restricted in some suitable function spaces, via the local linking method introduced by Shujie Li–Jiaquan Liu [22, 24] (see also [14, 23, 41]). The estimates of the components (u,v) in different parts of the function space 𝐸 are playing crucial roles in solving this problem.

For (u,v) that belongs to the orthogonal complement of ker⁡L×ker⁡L in 𝐸, we can control these components by some compact embedding estimate (2.9).
For (u,v)∈ker⁡L×ker⁡L, the compact properties will be lost. To overcome this difficulty, we apply the monotonicity technique to analyze the behavior of the nonlinear terms more precisely, as the linear coupling constant 𝜀 is sufficiently small. We require condition 4 of Theorem 1.1 to obtain this goal.

Subsequently, we study the asymptotic properties of the solutions constructed in Theorem 1.1 as ε→0. We establish Lemma 4.1 to obtain the uniform bound of the solutions (uε,vε)∈E to (1.1) for any |ε|<ε0, which leads to the strong convergence of (uε,vε)→(U0,V0) in L2⁢(Ω)×L2⁢(Ω) as ε→0. Then we verify that U0,V0 are weak solutions of (W1) and (W2) respectively, by a limiting argument with the aid of some precise energy estimates (4.9)–(4.13) to approach it.

Finally, we will improve the regularity of the weak solutions. We carry out the proof by two steps.

With the help of the presentation theorem for periodic solutions to the scalar wave equations [5, 29], we can use some a-priori estimates and comparison methods to achieve the L∞-estimate of the solutions (u,v) for (1.1) constructed by Theorem 1.1. The proof depends on the linear coupling structure of the system. See Section 5 for details.
Relying on the nature of the nonlinearities and the above L∞-estimate, we can generalize the continuity method [6, 29] used in the scalar wave equations to the case of system (1.1) taking account of the linear coupling effect. For sufficiently small 𝜀, the integral estimates in [6] are improved to prove the higher regularity of the solutions (see Section 6).

We organize the paper as follows. In Section 2, we give the functional scheme and define a suitable function space 𝐸 to work in it, with the aid of Fourier expansion formulations for the functions that satisfy (1.1b) and (1.1c). Then we introduce a decomposition of 𝐸 and prepare some basic embedding properties, which enable us to solve (1.1) conveniently. Section 3 is devoted to proving Theorem 1.1 via the local linking method. One of the major ingredients in the proof is to verify the (PS)^∗ condition by a compact argument together with a monotonicity technique (see Lemma 3.1 and Lemma 3.2). Then, in Section 4, we investigate the limit behavior of the solutions for (1.1) and prove Theorem 1.3. At last, we turn to study the further regularity properties of the time-periodic solution for (1.1) and prove Theorem 1.4, Theorem 1.5 in Section 5 and Section 6 respectively.

2 The Variational Framework

In this section, we present the variational framework which will be used to solve system (1.1). First, we define the energy functional and its working space as follows.

2.1 Functional Setting

Using the Fourier series, the solutions to the linear equation

ϕt⁢t-ϕx⁢x=h⁢(t,x),0<t<2⁢π, 0<x<π,

with conditions of ϕ⁢(t,0)=ϕ⁢(t,π)=0 and ϕ⁢(t+2⁢π,x)=ϕ⁢(t,x) have a expansion of the form

ϕ⁢(t,x)=∑j∈Z+,k∈Zaj⁢k⁢sin⁡(j⁢x)⁢ei⁢k⁢t,where⁢aj⁢k¯=aj,-k.

Then, for u⁢(t,x)=∑j∈Z+,k∈Zuj⁢k⁢sin⁡(j⁢x)⁢ei⁢k⁢t and v=∑j∈Z+,k∈Zvj⁢k⁢sin⁡(j⁢x)⁢ei⁢k⁢t, the inner product in L2⁢(Ω) can be formulated by

⟨u,v⟩=∬Ωu⁢(t,x)⁢v⁢(t,x)¯⁢dt⁢dx=π2⁢∑j∈Z+,k∈Zuj⁢k⁢vj⁢k¯,

and we can write the quadratic form as

⟨L⁢u,v⟩=∬Ω(∂2⁡u∂⁡t2-∂2⁡u∂⁡x2)⁢v¯⁢dt⁢dx=π2⁢∑j∈Z+,k∈Z(j2-k2)⁢uj⁢k⁢vj⁢k¯.

Motivated by [14], it is natural to introduce the Hilbert spaces

(2.1)H={u∈L2⁢(Ω):∥u∥H2=π2⁢∑j∈Z+,k∈Zj≠|k||j2-k2+b|⁢|uj⁢k|2+π2⁢∑j∈Z+,k∈Zj=|k||uj⁢k|2<∞}

and E=H×H as our working spaces, where 𝐸 is equipped with the norm ∥(u,v)∥E=(∥u∥H2+∥v∥H2)1/2.

For (u,v)∈E=H×H, let

(2.2)Φ⁢(u,v)=-12⁢⟨L⁢u,u⟩-b2⁢∬Ωu2⁢dt⁢dx-12⁢⟨L⁢v,v⟩-b2⁢∬Ωv2⁢dt⁢dx-ε⁢∬Ωu⁢v⁢dt⁢dx-∬ΩF⁢(t,x,u)⁢dt⁢dx-∬ΩG⁢(t,x,v)⁢dt⁢dx.

In the rest of this paper, we denote ∫⋅=∬Ω⋅dtdx for convenience.

Thus Φ is a C1 functional on 𝐸, and the Gateaux derivative of Φ is

(2.3)⟨Φ′⁢(u,v),(φ,ψ)⟩=-⟨L⁢u,φ⟩-b⁢∫u⁢φ-ε⁢∫v⁢φ-⟨L⁢v,ψ⟩-b⁢∫v⁢ψ-ε⁢∫u⁢ψ-∫f⁢(t,x,u)⁢φ-∫g⁢(t,x,v)⁢ψ for all⁢(u,v),(φ,ψ)∈E.

Then (u,v) is a weak solution of system (1.1) if and only if Φ′⁢(u,v)=0.

2.2 Local Linking Structure

The notion of local linking introduced by S. J. Li and M. Willem [23] is a powerful tool to study the existence of critical points for strongly indefinite functionals.

Definition 2.1

Let 𝐸 be a Banach space, and E=E1⊕E2 is a direct sum decomposition of 𝐸 (noting that both of E1 and E2 may be infinite-dimensional spaces). Then Φ∈C1⁢(E,R) is said to have a local linking at 0 if, for some r>0,

Φ⁢(u)≥0for⁢u∈E1,∥u∥≤r,

Φ⁢(u)≤0for⁢u∈E2,∥u∥≤r.

In the case of dim⁡E1=dim⁡E2=∞, it is necessary to explore the Galerkin approximation method and some compactness argument to construct the critical points of the functional Φ in 𝐸. To this end, we need the following compactness condition which generalizes the (PS) condition.

Suppose that E11⊂E21⊂⋯⊂E1, E12⊂E22⊂⋯⊂E2 are two sequences of finite-dimensional subspaces such that

Ej=⋃n∈NEnj¯,j=1,2.

For two multi-indices θ=(θ1,θ2) and β=(β1,β2)∈N2, we denote θ≤β if θ1≤β1, θ2≤β2. A sequence (θn)⊂N2 is said to be admissible if, for each θ∈N2, there is an m∈N such that θn≥θ for all n≥m. For θ=(θ1,θ2), let Eθ=Eθ11⊕Eθ22 and Φθ=Φ|Eθ.

Definition 2.2

The functional Φ∈C1⁢(E,R) is said to satisfy the condition (PS)^∗ if every sequence (uθn) with (θn)⊂N2 being admissible such that

uθn∈Eθn,supn⁡Φ⁢(uθn)<∞ and Φθn′⁢(uθn)→0 as⁢n→∞

contains a subsequence converging to a critical point of Φ.

We will use the following abstract proposition to solve system (1.1).

Proposition A

Proposition A ([23, 41])

Suppose that Φ∈C1⁢(E,R) and

Φ satisfies (PS)* condition,
Φ has a local linking at 0,
Φ maps bounded sets into bounded sets,
for every m∈N, Φ⁢(u)→-∞ as ∥u∥→∞, u∈Em1⊕E2.

Then Φ has a nontrivial critical point u0 in 𝐸.

Remark 2.3

In [14, 23], it is also pointed out that the critical value corresponding to the critical point u0 obtained in Proposition A satisfies

Φ⁢(u0)≤c,where⁢c=supu∈Em1+11⊕E2⁡Φ⁢(u),and⁢m1⁢is a positive integer.

In order to apply Proposition A to the functional defined by (2.2), we shall introduce the direct sum decomposition of the Banach space E=H×H, where 𝐻 occurs in (2.1). Some notation is defined as follows:

Hb+ is the subspace which is spanned by the functions sin⁡(j⁢x)⁢ei⁢k⁢t, where j∈Z+, k∈Z satisfy j2-k2>-b and j≠|k|;
Hb- is the subspace which is spanned by the functions sin⁡(j⁢x)⁢ei⁢k⁢t, where j2-k2<-b;
H0≡ker⁡L is the subspace which is spanned by the functions sin⁡(j⁢x)⁢ei⁢k⁢t for j=|k|;
Eb+=Hb+×Hb+, Eb-=Hb-×Hb- and E0=H0×H0;
E1=Eb-, E2=Eb+⊕E0, and we see dim⁡Ej=∞ for j=1, 2;
Emj=span⁡{e1j,…,emj}, where (enj)n=1∞ is a basis for Ej, j=1,2.

We have E=E1⊕E2. Moreover, Em1,Em2 are finite-dimensional spaces for every m∈Z, and E11⊂E21⊂⋯⊂E1, E12⊂E22⊂⋯⊂E2.

2.3 Basic Estimates

At the end of this section, we list some basic formulas and properties of the Banach spaces on which we will work. For r≥1, we denote by Lr⁢(Ω) the space of functions u⁢(t,x) with the norm

∥u∥Lr=(∬Ω|u⁢(t,x)|r⁢dt⁢dx)1/r.

Formulas of the Norms ∥⋅∥L2, ∥⋅∥H and Functional Φ

For u,v∈H and

u=∑j∈Z+k∈Zuj⁢k⁢sin⁡(j⁢x)⁢ei⁢k⁢t,v=∑j∈Z+k∈Zvj⁢k⁢sin⁡(j⁢x)⁢ei⁢k⁢t,

let u=u++u-+y, v=v++v-+z, where u+,v+∈Hb+, u-,v-∈Hb- and y,z∈H0. By the orthogonality of the subspaces Hb+, Hb- and H0, we can write the inner product in L2⁢(Ω) as ⟨u,v⟩=⟨u+,v+⟩+⟨u-,v-⟩+⟨y,z⟩, and

∥u∥L22=∥u+∥L22+∥u-∥L22+∥y∥L22=π2⁢∑j∈Z+k∈Z|uj⁢k|2.

Noting that -b∉σ⁢(L) and L⁢y=0 for y∈H0, we have ⟨L⁢u,v⟩=⟨L⁢u+,v+⟩+⟨L⁢u-,v-⟩ and

(2.4)⟨(L+b)⁢u,u⟩=π2⁢∑j∈Z+k∈Z(j2-k2+b)⁢|uj⁢k|2=∥u+∥H2-∥u-∥H2+b⁢∥y∥L22.

With the aid of (2.4), we can formulate the energy functional (2.2) as

(2.5)Φ⁢(u,v)=-12⁢∥u+∥H2+12⁢∥u-∥H2-b2⁢∥y∥L22-12⁢∥v+∥H2+12⁢∥v-∥H2-b2⁢∥z∥L22-ε⁢∫u⁢v-∫F⁢(t,x,u)-∫G⁢(t,x,v).

Some Embedding Estimates

In view of (1.2), there is η>0 such that

(2.6)∥u+∥H2=π2⁢∑j2-k2>-bj≠|k|(j2-k2+b)⁢|uj⁢k|2≥η⁢∥u+∥L22 and ∥u-∥H2≥η⁢∥u-∥L22.

Thus, by (2.1) and (2.6), we have

(2.7)∥u∥L22≤κ⁢∥u∥H2,where⁢κ=max⁡{1/η,1}.

The following properties are well known (see [14] for instance):

(2.8)∥w∥Lr≤C⁢∥w∥H for⁢w∈Hb+⊕Hb-⁢and⁢r≥1,

where C>0 is a constant which only depends on 𝑟. Furthermore, the embedding

(2.9)Hb+⊕Hb-↪Lr⁢(Ω)⁢is compact for⁢r≥1.

Remark 2.4

Let us point out that we lose the compact embedding from H0 to Lr⁢(Ω) for r>2 because of 0∈σ⁢(L) and dim⁡ker⁡L=∞. Moreover, according to the definition of 𝐻 (see (2.1)), we do not obtain that ∥w∥Lr≤C⁢∥w∥H for any w∈H and r>2. Thus we need careful computations to study the behavior of the null components and nonlinear terms appearing in the functional Φ.

3 Existence of Weak Solutions

In this section, we prove Theorem 1.1. To achieve this goal, we will check that the conditions of Proposition A hold for the functional Φ defined by (2.2). We use 𝐶 and c*,d* with quantity subscripts to stand for different constants in the rest of this article.

Verification of 1

Let (uθn,vθn)∈Eθn:=Eθn11⊕Eθn22, where θn=(θn1,θn2)∈N2, the sequence {θn}n=1∞ is admissible, and Eθn11, Eθn22 are finite-dimensional spaces defined in Subsection 2.2.

We suppose {(uθn,vθn)}n=1∞ is a (PS)^∗ sequence of Φ, that is

d:=supn⁡Φ⁢(uθn,vθn)<∞ and Φθn′⁢(uθn,vθn)→0 as⁢n→∞,

where Φθn′=(Φ|Eθn)′. First, under the assumptions of Theorem 1.1, we show the following lemma.

Lemma 3.1

Any (PS)* sequence is bounded.

Proof

For simplicity, we denote (u,v)=(uθn,vθn).

Step 1. Estimates of ∥u∥Lp+1 and ∥v∥Lq+1. By assumption 3 and (1.3), a direct calculation shows

Φ⁢(u,v)-12⁢⟨Φθn′⁢(u,v),(u,v)⟩=12⁢∫f⁢(t,x,u)⁢u-∫F⁢(t,x,u)+12⁢∫g⁢(t,x,v)⁢v-∫G⁢(t,x,v)≥p-12⁢∫F⁢(t,x,u)+q-12⁢∫G⁢(t,x,v)≥c1⁢∥u∥Lp+1p+1+c1⁢∥v∥Lq+1q+1-c2.

Since {(u,v)}={(uθn,vθn)} is a (PS)^∗ sequence of Φ, for 𝑛 large enough, we have

c1⁢∥u∥Lp+1p+1+c1⁢∥v∥Lq+1q+1-c2≤d+∥(u,v)∥E.

Hence there is c3>0 such that

(3.1)∥u∥Lp+1≤c3+c3⁢∥(u,v)∥E1p+1 and ∥v∥Lq+1≤c3+c3⁢∥(u,v)∥E1q+1.

Step 2. Estimates of ∥u±∥H, ∥v±∥H, ∥y∥L2 and ∥z∥L2. We decompose u=u++u-+y, v=v++v-+z, where u+,v+∈Hb+, u-,v-∈Hb- and y,z∈H0. Noting that ∥u∥L22=∥u+∥L22+∥u-∥L22+∥y∥L22, then by virtue of p,q>1 and (3.1), we estimate

(3.2)∥y∥L22≤∥u∥L22≤c⁢∥u∥Lp+12≤c4+c4⁢∥(u,v)∥E2p+1,

(3.3)∥z∥L22≤∥v∥L22≤c⁢∥v∥Lq+12≤c4+c4⁢∥(u,v)∥E2q+1.

Taking (φ,ψ)=(u+,v+) in (2.3), from the orthogonality of the subspaces Hb+, Hb- and H0, we get

⟨Φθn′⁢(u,v),(u+,v+)⟩=-⟨(L+b)⁢u+,u+⟩-⟨(L+b)⁢v+,v+⟩-2⁢ε⁢∫u+⁢v+-∫f⁢(t,x,u)⁢u+-∫g⁢(t,x,v)⁢v+.

In fact, of ∥u+∥H2=⟨(L+b)⁢u+,u+⟩ and ∥v+∥H2=⟨(L+b)⁢v+,v+⟩ for u+,v+∈Hb+, when 𝑛 is large enough, we obtain

(3.4)∥u+∥H2+∥v+∥H2≤o⁢(1)-2⁢ε⁢∫u+⁢v+-∫f⁢(t,x,u)⁢u+-∫g⁢(t,x,v)⁢v+.

An argument similar to (3.2) provides that

(3.5)-2⁢ε⁢∫u+⁢v+≤2⁢|ε|⁢∥u+∥L2⁢∥v+∥L2≤c⁢∥u∥Lp+1⁢∥v∥Lq+1≤c5+c5⁢∥(u,v)∥E1p+1+c5⁢∥(u,v)∥E1q+1+c5⁢∥(u,v)∥E1p+1+1q+1.

By assumption 1, (3.1) and the Hölder inequality, we get

(3.6)-∫f⁢(t,x,u)⁢u+≤c0⁢∫(|u+|+|u|p⁢|u+|)≤c⁢∥u+∥L2+c0⁢∥u∥Lp+1p⁢∥u+∥Lp+1≤c6⁢(1+∥(u,v)∥Epp+1)⁢∥u+∥H,

where the last inequality is deduced by (2.7), (2.8) and (3.1). Similarly, we have

(3.7)-∫g⁢(t,x,v)⁢v+≤c⁢∥v+∥L2+c0⁢∥v∥Lq+1q⁢∥v+∥Lq+1≤c6⁢(1+∥(u,v)∥Eqq+1)⁢∥v+∥H.

Inserting (3.5)–(3.7) into the right-hand side of (3.4), and by the inequalities

∥u+∥H≤∥(u,v)∥E and ∥v+∥H≤∥(u,v)∥E,

we know

(3.8)∥u+∥H2+∥v+∥H2≤o⁢(1)+c5+c5⁢∥(u,v)∥E1p+1+c5⁢∥(u,v)∥E1q+1+c5⁢∥(u,v)∥E1p+1+1q+1+c6⁢(1+∥(u,v)∥Epp+1+∥(u,v)∥Eqq+1)⁢∥(u,v)∥E.

For u-,v-∈Hb-, analogue to (3.8), we also derive

(3.9)∥u-∥H2+∥v-∥H2=⟨Φθn′⁢(u,v),(u-,v-)⟩+2⁢ε⁢∫u-⁢v-+∫f⁢(t,x,u)⁢u-+∫g⁢(t,x,v)⁢v-≤c5′+c5′⁢∥(u,v)∥E1p+1+c5′⁢∥(u,v)∥E1q+1+c5′⁢∥(u,v)∥E1p+1+1q+1+c6′⁢(1+∥(u,v)∥Epp+1+∥(u,v)∥Eqq+1)⁢∥(u,v)∥E.

Step 3. Bound of ∥(u,v)∥E. Observing that

∥(u,v)∥E2=∥u+∥H2+∥v+∥H2+∥u-∥H2+∥v-∥H2+∥y∥L22+∥z∥L22

and using the estimates of (3.2), (3.3), (3.8), (3.9), we arrive at

∥(u,v)∥E2≤c7+c7⁢∥(u,v)∥E1p+1+c7⁢∥(u,v)∥E1q+1+c7⁢∥(u,v)∥E1p+1+1q+1+c7⁢∥(u,v)∥E2p+1+c7⁢∥(u,v)∥E2q+1+c7⁢∥(u,v)∥E+c7⁢∥(u,v)∥Epp+1+1+c7⁢∥(u,v)∥Eqq+1+1.

In view of p,q>1, we find all the powers of ∥(u,v)∥E in the right-hand side of the preceding inequality are less than 2. Hence there exists M>0 which is independent of 𝑛 such that ∥(u,v)∥E≤M, and we conclude that any (PS)^∗ sequence {(uθn,vθn)} of Φ is bounded. ∎

In what follows, under the assumptions of Theorem 1.1, we assert that the functional Φ satisfies the (PS)^∗ condition.

Lemma 3.2

Let {(uθn,vθn)} be a (PS)* sequence of Φ. Then {(uθn,vθn)} contains a subsequence which converges to a critical point of Φ.

Proof

Since 𝐸 is a Hilbert space, Lemma 3.1 guarantees that {(uθn,vθn)} converge weakly to some (u,v)∈E along with a subsequence. We decompose

uθn=uθn++uθn-+yθn,vθn=vθn++vθn-+zθn,

where (uθn+,vθn+)∈Eb+, (uθn-,vθn-)∈Eb- and (yθn,zθn)∈E0. Let (u+,v+)∈Eb+, (u-,v-)∈Eb- and (y,z)∈E0 be the weak limits of {(uθn+,vθn+)}, {(uθn-,vθn-)} and {(yθn,zθn)} respectively.

We will conclude that {(uθn,vθn)} converge strongly to (u,v)=(u++u-+y,v++v-+z), by extracting a subsequence if necessary.

Strong convergence of {(uθn+,vθn+)} and {(uθn-,vθn-)} in 𝐸. For (uθn+,vθn+)∈Eb+ and (u+,v+)∈Eb+, by virtue of the weak convergence of uθn+⇀u+ and vθn+⇀v+ in 𝐻, we have

⟨(L+b)⁢u+,uθn+-u+⟩→0 and ⟨(L+b)⁢v+,vθn+-v+⟩→0 as⁢n→∞.

Hence it follows that, for large 𝑛,

(3.10)∥uθn+-u+∥H2+∥vθn+-v+∥H2=⟨(L+b)⁢(uθn+-u+),uθn+-u+⟩+⟨(L+b)⁢(vθn+-v+),vθn+-v+⟩=⟨(L+b)⁢uθn+,uθn+-u+⟩+⟨(L+b)⁢vθn+,vθn+-v+⟩+o⁢(1).

From (2.3), the first two terms in the right-hand side of (3.10) can be expressed by

(3.11)⟨(L+b)⁢uθn+,uθn+-u+⟩+⟨(L+b)⁢vθn+,vθn+-v+⟩=-⟨Φ′⁢(uθn,vθn),(uθn+-u+,vθn+-v+)⟩-∫f⁢(t,x,uθn)⁢(uθn+-u+)-ε⁢∫vθn⁢(uθn+-u+)-ε⁢∫uθn⁢(vθn+-v+)-∫g⁢(t,x,vθn)⁢(vθn+-v+).

To control (3.11), we denote by Pθn the projection operator from 𝐸 to its subspace Eθn, and we represent the first term in the right-hand side of (3.11) as

⟨Φ′⁢(uθn,vθn),(uθn+-u+,vθn+-v+)⟩=⟨Φθn′⁢(uθn,vθn),(uθn+-Pθn⁢u+,vθn+-Pθn⁢v+)⟩-⟨Φ′⁢(uθn,vθn),(I-Pθn)⁢(u+,v+)⟩.

Noting that (uθn+-Pθn⁢u+,vθn+-Pθn⁢v+)∈Eθn, then the facts that Φθn′⁢(uθn,vθn)→0 and (I-Pθn)⁢(u+,v+)→0 assure that

(3.12)⟨Φ′⁢(uθn,vθn),(uθn+-u+,vθn+-v+)⟩→0 as⁢n→∞.

Observe that Hb+⊕Hb- embeds compactly into Lr⁢(Ω) for r≥1. Then uθn+⇀u+, vθn+⇀v+ in 𝐻 imply that uθn+→u+, vθn+→v+ strongly in Lr⁢(Ω) for every r≥1. Moreover, since {uθn}, {vθn} are bounded in 𝐻 and from (2.7), we obtain

(3.13)|ε⁢∫vθn⁢(uθn+-u+)|≤|ε|⁢∥vθn∥L2⁢∥uθn+-u+∥L2→0,

(3.14)|ε⁢∫uθn⁢(vθn+-v+)|≤|ε|⁢∥uθn∥L2⁢∥vθn+-v+∥L2→0

as n→∞. By assumption 1 together with the Hölder inequality, it follows from (3.1) and the boundedness of ∥(uθn,vθn)∥E that

(3.15)|∫f⁢(t,x,uθn)⁢(uθn+-u+)|≤c8⁢∥uθn+-u+∥L2+c8⁢∥uθn∥Lp+1p⁢∥uθn+-u+∥Lp+1→0,

(3.16)|∫g⁢(t,x,vθn)⁢(vθn+-v+)|≤c8⁢∥vθn+-v+∥L2+c8⁢∥vθn∥Lq+1q⁢∥vθn+-v+∥Lq+1→0

as n→∞. Collecting (3.10)–(3.16), we infer

∥uθn+-u+∥H2+∥vθn+-v+∥H2→0 as⁢n→∞,

which means {(uθn+,vθn+)} converge to (u+,v+) strongly in 𝐸. Furthermore, for (uθn-,vθn-)∈Eb- and (u-,v-)∈Eb-, a similar argument allows us to obtain (uθn-,vθn-)→(u-,v-) strongly in 𝐸.

To finish the proof of this lemma, it remains to derive the strong convergence of (yθn,zθn)→(y,z) in 𝐸, where (yθn,zθn),(y,z)∈E0. Keeping in mind that the embedding from E0 to Lr⁢(Ω) is not compact, thus the above procedure is invalid to control the components (y,z)∈E0. In order to deal with the difficulties stemming from the lack of compactness, we shall employ the monotonicity method to study the asymptotic behavior of {(yθn,zθn)}.

Strong convergence of {(yθn,zθn)} in 𝐸 for small 𝜀. Recalling (2.1), it suffices to show that ∥yθn-y∥L2→0, ∥zθn-z∥L2→0 as n→∞, where (y,z) is the weak limit of {(yθn,zθn)} in E0.

From (2.3) and setting (φ,ψ)=(yθn-y,zθn-z), then

⟨L⁢uθn,yθn-y⟩=⟨L⁢yθn,yθn-y⟩=0,⟨L⁢vθn,zθn-z⟩=⟨L⁢zθn,zθn-z⟩=0

imply that

(3.17)-⟨Φ′⁢(uθn,vθn),(yθn-y,zθn-z)⟩=∫f⁢(t,x,uθn)⁢(yθn-y)+∫g⁢(t,x,vθn)⁢(zθn-z)+b⁢∫uθn⁢(yθn-y)+ε⁢∫vθn⁢(yθn-y)+b⁢∫vθn⁢(zθn-z)+ε⁢∫uθn⁢(zθn-z).

Since yθn⇀y, zθn⇀z in L2⁢(Ω), and by the orthogonality of the subspaces Hb+, Hb- and H0, we conclude that

b⁢∫uθn⁢(yθn-y)+b⁢∫vθn⁢(zθn-z)=b⁢∫yθn⁢(yθn-y)+b⁢∫zθn⁢(zθn-z)=b⁢∫(yθn-y)2+b⁢∫(zθn-z)2+o⁢(1),

ε⁢∫vθn⁢(yθn-y)+ε⁢∫uθn⁢(zθn-z)=ε⁢∫zθn⁢(yθn-y)+ε⁢∫yθn⁢(zθn-z)=2⁢ε⁢∫(yθn-y)⁢(zθn-z)+o⁢(1) as⁢n→∞.

Moreover, an analogue of the argument in (3.12) gives ⟨Φ′⁢(uθn,vθn),(yθn-y,zθn-z)⟩→0 as n→∞.

Therefore, the previous three estimates and (3.17) allow us to deduce that

(3.18)(b-|ε|)⁢∥yθn-y∥L22+(b-|ε|)⁢∥zθn-z∥L22≤b⁢∥yθn-y∥L22+b⁢∥zθn-z∥L22+2⁢ε⁢∫(yθn-y)⁢(zθn-z)=-∫f⁢(t,x,uθn)⁢(yθn-y)-∫g⁢(t,x,vθn)⁢(zθn-z)+o⁢(1) as⁢n→∞.

To control the right-hand side of (3.18), we rewrite

(3.19)∫f⁢(t,x,uθn)⁢(yθn-y)=∫[f⁢(t,x,uθn++uθn-+yθn)-f⁢(t,x,uθn++uθn-+y)]⁢(yθn-y)+∫[f⁢(t,x,uθn++uθn-+y)-f⁢(t,x,u++u-+y)]⁢(yθn-y)+∫f⁢(t,x,u++u-+y)⁢(yθn-y):=I1+I2+I3.

We estimate I1, I2 and I3 in the sequel. Noting that f⁢(t,x,ξ) is nondecreasing in 𝜉 by condition 4 of Theorem 1.1, we have I1≥0 immediately.

To estimate I2 and I3, firstly, we check yθn⇀y weakly in Lp+1⁢(Ω). For uθn=uθn++uθn-+yθn, in view of (3.1) and the embedding Hb±↪Lp+1⁢(Ω), we get

(3.20)∥yθn∥Lp+1≤∥uθn∥Lp+1+∥uθn+∥Lp+1+∥uθn-∥Lp+1≤∥uθn∥Lp+1+c⁢∥uθn+∥H+c⁢∥uθn-∥H≤c3+c3⁢∥(uθn,vθn)∥E1p+1+2⁢c⁢∥(uθn,vθn)∥E.

Then the boundedness of the (PS)^∗ sequence {(uθn,vθn)} ensures that {yθn} is bounded in Lp+1⁢(Ω), and {yθn} possesses a subsequence which converge weakly in Lp+1⁢(Ω). Recording that Lp+1⁢(Ω)↪L2⁢(Ω) for p>1 and 𝑦 is the weak limit of {yθn} in L2⁢(Ω), hence by the uniqueness of weak limit, we have yθn⇀y weakly in Lp+1⁢(Ω), with a subsequence still renamed by {yθn}.

By condition 1 of Theorem 1.1, we know the operator f:ξ↦f⁢(t,x,ξ) is continuous from Lp+1⁢(Ω) to Lp+1p⁢(Ω). Since uθn+→u+ and uθn-→u- in 𝐻, we have uθn++uθn-→u++u- strongly in Lp+1⁢(Ω) via the embedding Hb+⊕Hb-↪Lp+1⁢(Ω). Therefore,

f⁢(t,x,uθn++uθn-+y)-f⁢(t,x,u++u-+y)→0 in⁢Lp+1p as⁢n→∞,

and f⁢(t,x,u++u-+y) is in Lp+1p⁢(Ω). By virtue of yθn-y∈Lp+1 and Lp+1p⁢(Ω)=(Lp+1⁢(Ω))*, we obtain that I2→0 and I3→0 as n→∞.

With the estimates of I1, I2, I3 in hand, then passing to the limit in (3.19) yields that

(3.21)∫f⁢(t,x,uθn)⁢(yθn-y)≥0 as⁢n→∞.

Moreover, with a similar computation, we find

∫g⁢(t,x,vθn)⁢(zθn-z)≥0 as⁢n→∞.

Hence, turning back to (3.18) and choosing |ε|<b, we have ∥yθn-y∥L22+∥zθn-z∥L22≤o⁢(1) as n→∞. Thereby, the proof of Lemma 3.2 is completed. ∎

Proof of 2

The following lemma shows the functional Φ satisfying the local linking structure for small 𝜀.

Lemma 3.3

Assume that 1–4 of Theorem 1.1 are satisfied; then, for 𝜀 sufficiently small, there exists ρ>0 such that

Φ⁢(u,v)≥0 for (u,v)∈E1∩Bρ,
Φ⁢(u,v)≤0 for (u,v)∈E2∩Bρ,

where E1=Eb-, E2=Eb+⊕E0 and Bρ={(u,v)∣∥(u,v)∥E≤ρ}.

Proof

(i) For (u,v)∈Eb-=Hb-×Hb-, by (2.4), we have

∥(u,v)∥E2=∥u∥H2+∥v∥H2=-⟨(L+b)⁢u,u⟩-⟨(L+b)⁢v,v⟩.

Thus the energy functional (2.5) can be represented by

Φ⁢(u,v)=12⁢∥(u,v)∥E2-ε⁢∫u⁢v-∫F⁢(t,x,u)-∫G⁢(t,x,v).

By (2.6), we get

ε⁢∫u⁢v≤|ε|2⁢∥u∥L22+|ε|2⁢∥v∥L22≤|ε|2⁢η⁢∥u∥H2+|ε|2⁢η⁢∥v∥H2=|ε|2⁢η⁢∥(u,v)∥E2.

Using (1.4) and the embedding Hb-↪Lr⁢(Ω) for r≥1, then for each ν>0, we deduce that

∫F⁢(t,x,u)+∫G⁢(t,x,v)≤ν⁢∥u∥L22+Cν⁢∥u∥Lp+1p+1+ν⁢∥v∥L22+Cν⁢∥v∥Lq+1q+1≤νη⁢∥(u,v)∥H2+c1⁢∥u∥Hp+1+c2⁢∥v∥Hq+1.

Putting ν=ε2, it holds that

Φ⁢(u,v)≥12⁢(1-2⁢|ε|η)⁢∥(u,v)∥E2-c1⁢∥u∥Hp+1-c2⁢∥v∥Hq+1.

Choosing |ε|<η2 and letting

ρ=min⁡{(η-2⁢|ε|2⁢c1⁢η)1p-1,(η-2⁢|ε|2⁢c2⁢η)1q-1},

then for (u,v)∈E1 and ∥(u,v)∥E≤ρ, we arrive at

Φ⁢(u,v)≥∥u∥H2⁢[12⁢(1-2⁢|ε|η)-c1⁢∥u∥Hp-1]+∥v∥H2⁢[12⁢(1-2⁢|ε|η)-c2⁢∥v∥Hq-1]≥0.

(ii) For (u,v)∈Eb+⊕E0, we split u=u++y, v=v++z, where u+,v+∈Hb+ and y,z∈H0. Now,

Φ⁢(u,v)=-12⁢⟨(L+b)⁢u+,u+⟩-12⁢⟨(L+b)⁢v+,v+⟩-b2⁢∫y2-b2⁢∫z2-ε⁢∫u⁢v-∫F⁢(t,x,u)-∫G⁢(t,x,v).

From (2.6), we get

-∫u⁢v≤12⁢(∥u+∥L22+∥y∥L22)+12⁢(∥v+∥L22+∥z∥L22)≤12⁢η⁢∥u+∥H2+12⁢∥y∥L22+12⁢η⁢∥v+∥H2+12⁢∥z∥L22.

Hence F⁢(t,x,u)≥0 and G⁢(t,x,v)≥0 lead to

Φ⁢(u,v)≤-12⁢(1-|ε|η)⁢∥u+∥H2-12⁢(1-|ε|η)⁢∥v+∥H2-12⁢(b-|ε|)⁢∥y∥L22-12⁢(b-|ε|)⁢∥z∥L22.

Then, for (u,v)∈E2, we have Φ⁢(u,v)≤0 by selecting |ε|<min⁡{η,b}. ∎

Proof of 3

Concerning the bound of Φ in a bounded set, we have the following lemma.

Lemma 3.4

Φ maps bounded sets into bounded sets.

Proof

Let R0>0 and D={(u,v)∈E∣∥(u,v)∥E≤R0}. We claim that there exists a constant M0>0 such that Φ⁢(u,v)≤M0 for each (u,v)∈D. In fact, we decompose u=u++u-+y, v=v++v-+z, where u+,v+∈Hb+, u-,v-∈Hb-, and y,z∈H0. Then, combining (2.5), (2.7) with the fact F⁢(t,x,u)≥0 and G⁢(t,x,v)≥0, we obtain

Φ⁢(u,v)≤12⁢∥u-∥H2+12⁢∥v-∥H2-ε⁢∫u⁢v≤12⁢∥u∥H2+12⁢∥v∥H2+|ε|2⁢∥u∥L22+|ε|2⁢∥v∥L22≤12⁢∥(u,v)∥E2+κ⁢|ε|2⁢∥u∥H2+κ⁢|ε|2⁢∥v∥H2≤1+κ⁢|ε|2⁢R02:=M0

for each (u,v)∈D. That is what we desire. ∎

Proof of 4

We move to verify that Φ fulfills the last condition of Proposition A.

Lemma 3.5

For every m∈N and (u,v)∈Em1⊕E2, we have Φ⁢(u,v)→-∞ as ∥(u,v)∥E→∞ and 𝜀 is sufficiently small.

Proof

Let u=u++u-+y, v=v++v-+z for (u+,v+)∈Eb+, (y,z)∈E0, and (u-,v-)∈Em1=span⁡{e11,…,em1}, where (en1)n=1∞ is a basis for E1=Eb-. With the aid of (2.6), the coupled term in (2.5) can be controlled by

-ε⁢∫u⁢v≤|ε|2⁢(∥u∥L22+∥v∥L22)≤|ε|2⁢η⁢(∥u+∥H2+∥u-∥H2+∥v+∥H2+∥v-∥H2)+|ε|2⁢(∥y∥L22+∥z∥L22).

To deal with the nonlinear forced terms in (2.5), we utilize (1.3) and the embeddings Lp+1⁢(Ω)↪L2⁢(Ω), Lq+1⁢(Ω)↪L2⁢(Ω) for p,q>1 to get

-∫F⁢(t,x,u)-∫G⁢(t,x,v)≤-c1⁢∥u∥Lp+1p+1-c1⁢∥v∥Lq+1q+1+c2≤-c3⁢∥u∥L2p+1-c3⁢∥v∥L2q+1+c2.

Noting the dimension of Em1 is finite and the norms in the function space Em1 are equivalent, then for (u-,v-) in Em1, ∥u-∥Hp+1≤c⁢∥u-∥L2p+1≤c⁢∥u∥L2p+1 and ∥v-∥Hq+1≤c⁢∥v∥L2q+1. Inserting the preceding estimates into (2.5), we have

Φ⁢(u,v)≤-12⁢(1-|ε|η)⁢(∥u+∥H2+∥v+∥H2)-12⁢(b-|ε|)⁢(∥y∥L22+∥z∥L22)+12⁢(1+|ε|η)⁢∥u-∥H2-c4⁢∥u-∥Hp+1+12⁢(1+|ε|η)⁢∥v-∥H2-c4⁢∥v-∥Hq+1+c2.

As ∥(u,v)∥E=(∥u∥H2+∥v∥H2)1/2→∞, then

∥u+∥H2+∥v+∥H2+∥y∥L22+∥z∥L22→∞ or
∥u-∥H2+∥v-∥H2→∞ holds.

If (i) is satisfied, then there exists C>0 such that

12⁢(1+|ε|η)⁢∥u-∥H2-c4⁢∥u-∥Hp+1+12⁢(1+|ε|η)⁢∥v-∥H2-c4⁢∥v-∥Hq+1≤C

for p,q>1. Hence it follows that Φ⁢(u,v)→-∞ by selecting

|ε|<min⁡{η,b} as⁢∥u+∥H2+∥v+∥H2+∥y∥L22+∥z∥L22→∞.

If (ii) holds, then

12⁢(1+|ε|η)⁢∥u-∥H2-c4⁢∥u-∥Hp+1+12⁢(1+|ε|η)⁢∥v-∥H2-c4⁢∥v-∥Hq+1→-∞

by virtue of p,q>1. We derive that Φ⁢(u,v)→-∞ as ∥u-∥H2+∥v-∥H2→∞ for |ε|<min⁡{η,b}. The conclusion of Lemma 3.5 is thereby obtained. ∎

Now, we have proved that the functional Φ∈C1⁢(E,R) satisfies conditions 1–4 of Proposition A, which ensure us to construct a nontrivial critical point (u,v) of Φ in 𝐸. Hence we finish the proof of Theorem 1.1.∎

4 Asymptotic Behavior of the Solutions as ε→0

In the following, we use ci and 𝐶 to denote positive constants which are independent of 𝜀, and whose value may differ from line to line.

Let (uε,vε) be the solution of (1.1) obtained in Theorem 1.1 for |ε|<ε0. We know

⟨(L+b)⁢uε,uε⟩+ε⁢∫uε⁢vε+∫f⁢(t,x,uε)⁢uε=0,

⟨(L+b)⁢vε,vε⟩+ε⁢∫uε⁢vε+∫g⁢(t,x,vε)⁢vε=0.

Then, by (2.2), (1.3) and 3 of Theorem 1.1, we have

(4.1)Φ⁢(uε,vε)=12⁢∫[f⁢(t,x,uε)⁢uε-F⁢(t,x,uε)]+12⁢∫[g⁢(t,x,vε)⁢vε-G⁢(t,x,vε)]≥p-12⁢∫F⁢(t,x,uε)+q-12⁢∫G⁢(t,x,vε)≥(p-1)⁢c12⁢∫|uε|p+1+(q-1)⁢c12⁢∫|vε|q+1-2⁢c2⁢π2.

On the other hand, we deduce from Remark 2.3 together with Lemma 3.4 and Lemma 3.5 that there exists a positive number c3 independent of 𝜀 such that

(4.2)Φ⁢(uε,vε)≤c3 for any⁢ε∈(-ε0,ε0).

Combining with (4.1), (4.2) and by virtue of p,q>1, we get

(4.3)∥uε∥Lp+1+∥vε∥Lq+1≤C for any⁢ε∈(-ε0,ε0).

Let εn∈(-ε0,ε0) be any sequence with εn→0 as n→∞. Subsequently, we will prove that (uεn,vεn) converge strongly to some (U0,V0) in L2⁢(Ω)×L2⁢(Ω) as n→∞, by passing to a subsequence, and justify that U0, V0 are weak solutions of the scalar wave equations (W1) and (W2) respectively.

At first, we decompose uεn=uεn++uεn-+yεn, vεn=vεn++vεn-+zεn, where (uεn+,vεn+)∈Eb+, (uεn-,vεn-)∈Eb- and (yεn,zεn)∈E0, and show the next lemma concerning the asymptotic behavior of (uεn,vεn).

Lemma 4.1

Passing to a subsequence of εn→0 as n→∞, we have

(uεn+,vεn+) converge strongly to some (U0+,V0+) in Eb+,
(uεn-,vεn-) converge strongly to some (U0-,V0-) in Eb-,
(yεn,zεn) converge strongly to some (U00,V00) in E0,
uεn⇀U0 weakly in Lp+1⁢(Ω), vεn⇀V0 weakly in Lq+1⁢(Ω), where U0=U0++U0-+U00, V0=V0++V0-+V00.

Proof

First, we establish the uniform bound for {(uεn,vεn)} in 𝐸. Recording (4.3) and the embedding properties of Lr⁢(Ω)↪L2⁢(Ω) for r≥2, we have

(4.4)∥yεn∥L22+∥zεn∥L22≤∥uεn∥L22+∥vεn∥L22≤c4⁢∥uεn∥Lp+12+c4⁢∥vεn∥Lq+12≤C.

For (uεn+,vεn+)∈Eb+, by 1, (2.8), (4.3) and the orthogonality of Hb+, Hb-, H0,

(4.5)∥uεn+∥H2+∥vεn+∥H2=⟨(L+b)⁢uεn,uεn+⟩+⟨(L+b)⁢vεn,vεn+⟩=-2⁢εn⁢∫uεn+⁢vεn+-∫f⁢(t,x,uεn)⁢uεn+-∫g⁢(t,x,vεn)⁢vεn+≤c5⁢(∥uεn∥L2⁢∥vεn∥L2+∥uεn∥L2+∥uεn∥Lp+1p⁢∥uεn+∥H+∥vεn∥L2+∥vεn∥Lq+1q⁢∥vεn+∥H)≤C⁢(1+∥(uεn+,vεn+)∥E).

Similarly, we get

(4.6)∥uεn-∥H2+∥vεn-∥H2=-⟨(L+b)⁢uεn,uεn-⟩-⟨(L+b)⁢vεn,vεn-⟩≤C⁢(1+∥(uεn-,vεn-)∥E) for⁢(uεn-,vεn-)∈Eb-.

Therefore, summing up (4.4)–(4.6), we arrive at ∥(uεn,vεn)∥E2≤C⁢(1+∥(uεn,vεn)∥E), which implies that

(4.7)∥(uεn,vεn)∥E≤C0,where⁢C0>0⁢is independent of⁢εn.

Then {(uεn,vεn)} converge weakly to some (U0,V0)∈E, by passing to a subsequence of εn→0 as n→∞. Split U0=U0++U0-+U00, V0=V0++V0-+V00, where (U0+,V0+)∈Eb+, (U0-,V0-)∈Eb- and (U00,V00)∈E0. We assert that (U0+,V0+), (U0-,V0-) and (U00,V00) satisfy (i), (ii), (iii) of this lemma.

Since Φ′⁢(uεn,vεn)=0, and by the orthogonality of Hb+, Hb-, H0, it follows that

∥uεn+-U0+∥H2=⟨(L+b)⁢(uεn+-U0+),uεn+-U0+⟩=⟨(L+b)⁢uεn,uεn+-U0+⟩-⟨(L+b)⁢U0,uεn+-U0+⟩=-εn⁢∫vεn⁢(uεn+-U0+)-∫f⁢(t,x,uεn)⁢(uεn+-U0+)-⟨(L+b)⁢U0,uεn+-U0+⟩.

By the weak convergence of uεn+⇀U0+ as n→∞, we can proceed as in the proof of (3.10), (3.13) and (3.15) in Lemma 3.2 to show that uεn+→U0+ strongly in Eb+, as n→∞. Then (i) holds. We can prove (ii) in the same way.

For yεn∈H0, zεn∈H0,

-⟨Φ′⁢(uεn,vεn),(yεn-U00,zεn-V00)⟩=⟨L⁢uεn,yεn-U00⟩+b⁢∫uεn⁢(yεn-U00)+εn⁢∫vεn⁢(yεn-U00)+∫f⁢(t,x,uεn)⁢(yεn-U00)+⟨L⁢vεn,zεn-V00⟩+b⁢∫vεn⁢(zεn-V00)+εn⁢∫uεn⁢(zεn-V00)+∫g⁢(t,x,vεn)⁢(zεn-V00).

By virtue of Φ′⁢(uεn,vεn)=0 and ⟨L⁢uεn,yεn-U00⟩=⟨L⁢vεn,zεn-V00⟩=0, we have

b⁢∫uεn⁢(yεn-U00)+εn⁢∫vεn⁢(yεn-U00)+b⁢∫vεn⁢(zεn-V00)+εn⁢∫uεn⁢(zεn-V00)=-∫f⁢(t,x,uεn)⁢(yεn-U00)-∫g⁢(t,x,vεn)⁢(zεn-V00).

By a proof analogous to (3.18)–(3.21) in Lemma 3.2, we deduce that

∥yεn-U00∥L22+∥zεn-V00∥L22→0 as⁢n→∞,

and (iii) is satisfied.

As a consequence of (i), (ii), (iii), we have {(uεn,vεn)} converge strongly to (U0,V0) in 𝐸. Particularly,

(4.8)(uεn,vεn)→(U0,V0) strongly in⁢L2⁢(Ω)×L2⁢(Ω),as⁢n→∞.

On the other hand, (4.3) implies that there exists some U1∈Lp+1⁢(Ω) such that {uεn} possesses a subsequence which converges weakly to U1 in Lp+1⁢(Ω). By virtue of Lp+1⁢(Ω)↪L2⁢(Ω) for p>1, and noting the fact that the weak limit of {uεn} is unique, we have U1=U0. Thus uεn⇀U0 weakly in Lp+1⁢(Ω). Similarly, we get vεn⇀V0 weakly in Lq+1⁢(Ω). Hence (iv) holds. ∎

Lemma 4.2

For any φ∈H∩Lp+1⁢(Ω), ψ∈H∩Lq+1⁢(Ω), then passing to a subsequence of εn→0 as n→∞, we have

(4.9)⟨(L+b)⁢uεn,uεn-φ⟩→⟨(L+b)⁢U0,U0-φ⟩,

(4.10)⟨(L+b)⁢vεn,vεn-ψ⟩→⟨(L+b)⁢V0,V0-ψ⟩,

(4.11)⟨f⁢(t,x,φ),uεn-φ⟩→⟨f⁢(t,x,φ),U0-φ⟩,

(4.12)⟨g⁢(t,x,ψ),vεn-ψ⟩→⟨g⁢(t,x,ψ),V0-ψ⟩,

(4.13)⟨εn⁢vεn,uεn-φ⟩→0 and ⟨εn⁢uεn,vεn-ψ⟩→0.

Proof

To reach (4.9), we write

⟨(L+b)⁢uεn,uεn-φ⟩-⟨(L+b)⁢U0,U0-φ⟩=⟨(L+b)⁢uεn,uεn-U0⟩+⟨(L+b)⁢(uεn-U0),U0-φ⟩:=A1+A2.

From the orthogonality of Hb+, Hb-, H0, it follows that

A1=⟨(L+b)⁢uεn+,uεn+-U0+⟩+⟨(L+b)⁢uεn-,uεn--U0-⟩+b⁢⟨yεn,yεn-U00⟩.

Then, by (4.7), Lemma 4.1 and using the Hölder inequality, we infer

|A1|≤∥uεn+∥H⁢∥uεn+-U0+∥H+∥uεn-∥H⁢∥uεn--U0-∥H+b⁢∥yεn∥L2⁢∥yεn-U00∥L2→0 as⁢n→∞.

Furthermore, since φ∈H, we deduce that A2→0 in the same way. Thus (4.9) holds, and we can obtain (4.10) similarly.

We come to prove (4.11). By condition 1 of Theorem 1.1, we have

(4.14)|⟨f⁢(t,x,φ),uεn-U0⟩|≤c0⁢∫|uεn-U0|+c0⁢∫|φ|p⁢|uεn-U0|.

Then (4.8), (iv) of Lemma 4.1 and |φ|p∈Lp+1p⁢(Ω)=(Lp+1⁢(Ω))* show the right-hand side of (4.14) goes to zero as n→∞, which gives (4.11). Furthermore, (4.12) also holds for any ψ∈H∩Lq+1⁢(Ω).

At last, by (4.7) and the Hölder inequality, it follows that

|⟨εn⁢vεn,uεn-φ⟩|+|⟨εn⁢uεn,vεn-ψ⟩|≤|εn|⁢∥vεn∥L2⁢∥uεn-φ∥L2+|εn|⁢∥uεn∥L2⁢∥vεn-ψ∥L2≤C⁢|εn|→0 as⁢n→∞.

Hence we arrive at (4.13). ∎

We are ready for the proof of Theorem 1.3.

Proof

It suffices to prove that

(4.15)⟨(L+b)⁢U0+f⁢(t,x,U0),ω⟩=0 ⁢for all⁢ω∈H∩Lp+1⁢(Ω),

(4.16)⟨(L+b)⁢V0+g⁢(t,x,V0),χ⟩=0 ⁢for all⁢χ∈H∩Lq+1⁢(Ω).

As (uεn,vεn) solves problem (1.1) with linear coupling constant ε=εn, then

(4.17)⟨(L+b)⁢uεn+f⁢(t,x,uεn)+εn⁢vεn,uεn-φ⟩=0 for all⁢φ∈H∩Lp+1⁢(Ω).

By 4 of Theorem 1.1, we know ⟨f⁢(t,x,uεn)-f⁢(t,x,φ),uεn-φ⟩≥0. Hence we derive from (4.17) that

(4.18)⟨(L+b)⁢uεn+f⁢(t,x,φ)+εn⁢vεn,uεn-φ⟩≤0 for all⁢φ∈H∩Lp+1⁢(Ω).

By (4.9), (4.11) and (4.13) of Lemma 4.2, then passing to the limit in (4.18), we have

(4.19)⟨(L+b)⁢U0+f⁢(t,x,φ),U0-φ⟩≤0 for all⁢φ∈H∩Lp+1⁢(Ω).

Choosing φ=U0-λ⁢ω with λ>0 in (4.19), then dividing by 𝜆 and letting λ→0, we obtain

⟨(L+b)⁢U0+f⁢(t,x,U0),ω⟩≤0,

and noting ω∈H∩Lp+1⁢(Ω) is chosen arbitrarily, we infer that U0 satisfies

⟨(L+b)⁢U0+f⁢(t,x,U0),ω⟩=0 for all⁢ω∈H∩Lp+1⁢(Ω).

Thereby, (4.15) is concluded. In the same manner, we obtain (4.16) by virtue of (4.10), (4.12) and (4.13). This concludes the proof of Theorem 1.3. ∎

5 Higher Regularity of the Solutions

In this section, we prove the solutions (u,v)∈E obtained in the previous section enjoy the higher regularity, provided that 𝜀 is sufficiently small.

5.1 A Representation Theorem for Single Wave Equation

To proceed, we collect some facts that are useful in improving the regularity of the weak solutions for (1.1).

Let Ω=[0,2⁢π]×[0,π], and ker⁡L is the kernel of the d’Alembert operator L=∂t2-∂x2. Denote by R⁢(L) the range of 𝐿. Then we have R⁢(L)=(ker⁡L)⊥. The following representation theorem plays an important role in the study of the regularity theory of scalar wave equations.

Proposition 5.1

Proposition 5.1 (see [5])

Given h∈L1⁢(Ω)∩R⁢(L) satisfying h⁢(t+2⁢π,x)=h⁢(t,x), then the solution of the wave equation

(W){L⁢w≡wt⁢t-wx⁢x=h⁢(t,x),(t,x)∈Ω,w⁢(t,0)=w⁢(t,π)=0,w⁢(t+2⁢π,x)=w⁢(t,x)

can be presented in the form w=w0+w1, where w0∈ker⁡L and w1∈(ker⁡L)⊥=R⁢(L). More precisely, we have w0⁢(t,x)=p⁢(t+x)-p⁢(t-x) for some p∈L1 which is 2⁢π-periodic and satisfying ∫02⁢πp⁢(τ)⁢dτ=0, and

w1⁢(t,x)=-12⁢∫xπdξ⁢∫t+x-ξt-x+ξh⁢(τ,ξ)⁢dτ+π-x2⁢π⁢∫0πdξ⁢∫t-ξt+ξh⁢(τ,ξ)⁢dτ.

Estimates of the Component w1 in R⁢(L)=(ker⁡L)⊥

From Proposition 5.1, the L∞-norm of w1∈R⁢(L) can be controlled by

(5.1)∥w1∥L∞≤c⁢∥h∥L1 for⁢h∈L1∩R⁢(L).

Furthermore, if h∈L∞⁢(Ω)∩R⁢(L), then we have

(5.2)∥w1∥C0,1≤c⁢∥h∥L∞,

where ∥⋅∥C0,1 is the norm of the Lipschitz space C0,1⁢(Ω).

An Integral Formula Concerning the Component in ker⁡L

If p⁢(s),q⁢(s) are L1 functions with period 2⁢π and satisfy ∫02⁢πp⁢(s)⁢ds=∫02⁢πq⁢(s)⁢ds=0, then a computation as in [28] shows that

∬Ωp⁢(t+x)⁢q⁢(t-x)⁢dt⁢dx=0.

We also require the next property to characterize the range of the operator𝐿: a function ℎ belongs to R⁢(L) if and only if

(5.3)∫0π[h⁢(t+x,x)-h⁢(t-x,x)]⁢dx=0.

In other words, the sufficient and necessary condition for the solvability of linear wave equation (W) is that the function h⁢(t,x) satisfies (5.3). See [5, 25] for more details.

5.2 Proof of Theorem 1.4

Let (u,v)∈E be a solution of (1.1) constructed by local linking method. We decompose it into

(u,v)=(u1+y,v1+z),

where u1,v1∈Hb+⊕Hb-≡R⁢(L) and y,z∈H0≡ker⁡L. We apply Proposition 5.1 to represent y,z by

y=p⁢(t+x)-p⁢(t-x) and z=q⁢(t+x)-q⁢(t-x)

for some 2⁢π-periodic functions p,q∈L1⁢([0,2⁢π]) such that ∫02⁢πp⁢(τ)⁢dτ=∫02⁢πq⁢(τ)⁢dτ=0.

The L∞-Regularity of the Components in Hb+⊕Hb-

Since (u,v) satisfy the system

{ut⁢t-ux⁢x+b⁢u+ε⁢v+f⁢(t,x,u)=0,vt⁢t-vx⁢x+b⁢v+ε⁢u+g⁢(t,x,v)=0, (t,x)∈Ω,

we shall derive the L∞- estimate of u1, v1∈Hb+⊕Hb- from Proposition 5.1.

Lemma 5.2

Assume that (u,v) is a solution of (1.1) obtained in Theorem 1.1, then there exists d1, d2>0, such that ∥u1∥L∞≤d1 and ∥v1∥L∞≤d2.

Proof

By (2.8), p>1 and 1 of Theorem 1.1, we have

∫|f⁢(t,x,u)|≤c0⁢∫(1+|u|p)≤c1⁢(1+∥u∥Ep)<∞,

which means f∈L1⁢(Ω×R). Noting that (2.8) also implies 𝑢, v∈L1⁢(Ω), then we can use estimate (5.1) to obtain a number d1>0 such that

(5.4)∥u1∥L∞≤c⁢∥b⁢u+ε⁢v+f⁢(t,x,u)∥L1≤d1<∞.

Similarly, we have

(5.5)∥v1∥L∞≤c⁢∥b⁢v+ε⁢u+g⁢(t,x,v)∥L1≤d2<∞

for some d2>0. ∎

The L∞-Regularity of the Components in H0≡ker⁡L

For the terms of y,z in H0, the proof of y,z∈L∞⁢(Ω) is a difficult task since the a-priori estimate (5.1) is invalid for 𝑦 and 𝑧. To this end, we set

(5.6)N1=∥p∥L∞⁢(Ω),N2=∥q∥L∞⁢(Ω).

Without loss of generality, we may suppose that there are s1 and s2 such that p⁢(s1)>N1-1 and q⁢(s2)>N2-1. We shall derive the upper bounds of N1 and N2.

Let f~⁢(t,x)=f⁢(t,x,u⁢(t,x)) and

(5.7)h1⁢(t,x)=b⁢u⁢(t,x)+ε⁢v⁢(t,x)+f~⁢(t,x),

where u⁢(t,x)=u1⁢(t,x)+p⁢(t+x)-p⁢(t-x), v⁢(t,x)=v1⁢(t,x)+q⁢(t+x)-q⁢(t-x).

We prepare the following two lemmas to prove Theorem 1.4.

Lemma 5.3

Assume that (u⁢(t,x),v⁢(t,x)) is a solution of (1.1) obtained in Theorem 1.1. Then there exists a number d3>0 such that

(5.8)∫0πf~⁢(s1-x,x)⁢dx≤d3-b⁢π⁢p⁢(s1)-2⁢ε⁢π⁢q⁢(s1).

Proof

Noting that L⁢u+h1⁢(t,x)=0, then putting t=s1 in (5.3), we have

(5.9)∫0πh1⁢(s1+x,x)⁢dx=∫0πh1⁢(s1-x,x)⁢dx.

Since p⁢(s1+2⁢x)-p⁢(s1)≤1 and u1⁢(s1+x,x)≤d1, it follows from 4, (5.4) and f∈C⁢(Ω×R,R) that

f~⁢(s1+x,x)≤f⁢(s1+x,x,d1+1)≤d4

for some d4>0. Therefore, by virtue of ∫0πq⁢(s1+2⁢x)⁢dx=0, we deduce from (5.4), (5.5) and the above inequality that

(5.10)∫0πh1⁢(s1+x,x)⁢dx≤∫0π[b⁢d1+|ε|⁢d2+ε⁢q⁢(s1+2⁢x)-ε⁢q⁢(s1)+d4]⁢dx≤d5-ε⁢π⁢q⁢(s1).

On the other hand, it follows from (5.4), (5.5) that

u1⁢(s1-x,x)≥-d1 and v1⁢(s1-x,x)≥-d2.

Furthermore, by the fact that ∫0πp⁢(s1-2⁢x)⁢dx=∫0πq⁢(s1-2⁢x)⁢dx=0, we have

(5.11)∫0πh1⁢(s1-x,x)⁢dx≥b⁢π⁢p⁢(s1)+ε⁢π⁢q⁢(s1)+∫0πf~⁢(s1-x,x)⁢dx-b⁢π⁢d1.

Hence, combining with (5.9)–(5.11), we arrive at (5.8). ∎

To proceed further, we denote

Λ1={x∈[0,π]:p⁢(s1-2⁢x)≤N12} and Λ2={x∈[0,π]:q⁢(s2-2⁢x)≤N22}.

Lemma 5.4

Assume that (u⁢(t,x),v⁢(t,x)) is a solution of (1.1) obtained in Theorem 1.1. Then meas⁡Λ1≥π3, meas⁡Λ2≥π3, and there exists a number d>0 such that

(5.12)∫Λ1f⁢(s1-x,x,N12-d1-1)⁢dx+∫Λ2g⁢(s2-x,x,N22-d2-1)⁢dx≤d.

Proof

(1) At first, we prove that meas⁡Λ1≥π3. In view of

p⁢(s1-2⁢x)≥{-N1for⁢x∈Λ1,12⁢N1for⁢x∈Λ1c,

we get

0=∫0πp⁢(s1-2⁢x)⁢dx=∫Λ1p⁢(s1-2⁢x)⁢dx+∫Λ1cp⁢(s1-2⁢x)⁢dx≥-N1⁢meas⁡Λ1+N12⁢(π-meas⁡Λ1),

which indicates meas⁡Λ1≥π3. Similarly, we also have meas⁡Λ2≥π3.

(2) Now we turn to the proof of (5.12). To this end, we represent the left-hand side of (5.8) in the form

(∫Λ1+∫Λ1c)⁢f~⁢(s1-x,x)⁢d⁢x:=Q~1+Q~2.

It follows from (5.6) that p⁢(s1)-p⁢(s1-2⁢x)≥12⁢N1-1 for x∈Λ1. Then, by 4 and (5.4), we know

Q~1≥∫Λ1f⁢(s1-x,x,N12-d1-1)⁢dx.

On the other hand, recording u1⁢(s1-x,x)+p⁢(s1)-p⁢(s1-2⁢x)≥-d1-1, then we deduce from 1 and 4 that

f~⁢(s1-x,x)≥f⁢(s1-x,x,-d1-1)≥-c0-c0⁢d1p,

which gives

Q~2≥-∫Λ1c(c0+c0⁢d1p)⁢dx≥-d6.

Thus, coming back to (5.8), we get

(5.13)∫Λ1f⁢(s1-x,x,N12-d1-1)⁢dx≤∫0πf~⁢(s1-x,x)⁢dx-Q~2≤d7-b⁢π⁢p⁢(s1)-2⁢ε⁢π⁢q⁢(s1).

Applying a similar procedure, we also have

(5.14)∫Λ2g⁢(s2-x,x,N22-d2-1)⁢dx≤d7′-b⁢π⁢q⁢(s2)-2⁢ε⁢π⁢p⁢(s2).

Choosing |ε|<b2, we infer from (5.6) that |p⁢(s2)|≤p⁢(s1)+1, |q⁢(s1)|≤q⁢(s2)+1, which assure

b⁢π⁢p⁢(s1)+2⁢ε⁢π⁢p⁢(s2)≥-2⁢ε⁢π and b⁢π⁢q⁢(s2)+2⁢ε⁢π⁢q⁢(s1)≥-2⁢ε⁢π.

Then, adding up (5.13) and (5.14), we obtain (5.12). ∎

Now we are ready to prove Theorem 1.4.

Proof of Theorem 1.4

Without loss of generality, we may assume that N1>2⁢d1+2 and N2>2⁢d2+2. Thus, from 2 and 4, we derive

∫Λ1f⁢(s1-x,x,N12-d1-1)⁢dx≥0,∫Λ2g⁢(s2-x,x,N22-d2-1)⁢dx≥0.

Now, we claim that N1<+∞. If not, from the fact that f⁢(s1-x,x,ξ)→+∞ as ξ→+∞ for x∈Λ1, we deduce that, for any γ>0, there are β>0 and a subset 𝐷 of Λ1 with meas⁡D<π6 such that

f⁢(s1-x,x,β)≥γ for⁢x∈Dc.

Hence, for N1>2⁢(β+d1), we infer from the above inequality and 4 that

(5.15)∫Λ1∩Dcf⁢(s1-x,x,N12-d1-1)⁢dx≥γ⁢meas⁡(Λ1∩Dc)≥16⁢γ⁢π.

On the other hand, by 2, 4 and N1>2⁢d1+2, we have

(5.16)∫Λ1∩Df⁢(s1-x,x,N12-d1-1)⁢dx≥0.

Then, with the aid of (5.12), (5.15) and (5.16), we show that

16⁢γ⁢π≤∫Λ1f⁢(s1-x,x,N12-d1-1)⁢dx≤d,

which is impossible when we fix γ>6⁢dπ. Therefore, we conclude that 𝑦 is essentially bounded. With a similar argument, we have z∈L∞. The proof of Theorem 1.4 is thereby completed. ∎

6 Proof of Theorem 1.5

This section is devoted to the proof of the continuity of the solutions for (1.1).

Let (u,v)∈L∞⁢(Ω)×L∞⁢(Ω) be a solution of (1.1) provided by Theorem 1.4. Split it into

(u,v)=(u1+y,v1+z),

with u1,v1∈Hb+⊕Hb-≡(ker⁡L)⊥ and y,z∈H0≡ker⁡L.

6.1 Continuity of the Regular Terms

It follows from 1 and the fact that u,v∈L∞ that b⁢u+ε⁢v+f⁢(t,x,u)∈L∞ and b⁢v+ε⁢u+g⁢(t,x,v)∈L∞. Hence, noting that (u,v)=(u1+y,v1+z) solves system (1.1), we infer from (5.2) that u1,v1∈C0,1⁢(Ω), which means

(6.1)|u1⁢(t,x)-u1⁢(τ,ζ)|+|v1⁢(t,x)-v1⁢(τ,ζ)|≤C⁢(|t-τ|+|x-ζ|)

for all (t,x),(τ,ζ)∈Ω. Consequently, we achieve that u1⁢(t,x) and v1⁢(t,x) are continuous on Ω.

6.2 Continuity of the Null Terms

We turn to prove that y,z∈C⁢(Ω). Noting that y,z∈ker⁡L can be represented in the form y=p⁢(t+x)-p⁢(t-x) and z=q⁢(t+x)-q⁢(t-x), where p,q∈L1⁢([0,2⁢π]) are 2⁢π-periodic and satisfy ∫02⁢πp⁢(τ)⁢dτ=∫02⁢πq⁢(τ)⁢dτ=0, then it suffices to show that 𝑝 and 𝑞 are continuous, which means

p⁢(t+h)-p⁢(t)→0,q⁢(t+h)-q⁢(t)→0 as⁢h→0 for a.e.⁢t∈[0,2⁢π].

To reach our goal, we denote p^h⁢(t)=p⁢(t+h)-p⁢(t), q^h⁢(t)=q⁢(t+h)-q⁢(t) for fixed |h|<14, and let

M1=∥p^h∥L∞⁢(Ω),M2=∥q^h∥L∞⁢(Ω).

Then M1≤2⁢N1<∞ and M2≤2⁢N2<∞, where N1=∥p∥L∞⁢(Ω) and N2=∥q∥L∞⁢(Ω). Moreover, without loss of generality, we may assume that there exist s1 and s2 such that

(6.2)p^h⁢(s1)>M1⁢(1-|h|),q^h⁢(s2)>M2⁢(1-|h|).

We intend to prove M1→0, M2→0 as |h|→0.

For simplicity, we denote p^⁢(t)=p^h⁢(t) and q^⁢(t)=q^h⁢(t). The following notations and estimates are in order. Define

ϕ⁢(τ)=min(t,x)∈Ω|s|≤2⁢N1⁡[f⁢(t,x,u1⁢(t,x)+s+τ)-f⁢(t,x,u1⁢(t,x)+s)],

ψ⁢(τ)=min(t,x)∈Ω|s|≤2⁢N2⁡[g⁢(t,x,v1⁢(t,x)+s+τ)-g⁢(t,x,v1⁢(t,x)+s)].

We shall prove that

(6.3)∫0π[ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))+ψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))]⁢dx≤C⁢|h|,

(6.4)∫Σ1ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))⁢dx+∫Σ2ψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))⁢dx≤C⁢|h|,

where C>0 is a number independent of ℎ, and

Σ1={x∈[0,π]:p^⁢(s1)-p^⁢(s1-2⁢x)≥M12},Σ2={x∈[0,π]:q^⁢(s2)-q^⁢(s2-2⁢x)≥M22}.

6.2.1 Proof of (6.3)

We carry out the argument in three steps.

Step 1

We establish the following lemma to study the behavior of the integral

J=∫0π[bp^(s1)-bp^(s1-2x)+εq^(s1)-εq^(s1-2x)-f~(s1-x,x)+f(s1-x,x,u1(s1-x,x)+p(s1+h)-p(s1+h-2x))]dx.

Lemma 6.1

Assume that (u⁢(t,x),v⁢(t,x)) is a solution of (1.1) obtained in Theorem 1.1. Then there exists a number C>0 independent of ℎ such that

(6.5)J≤C⁢|h|-ε⁢π⁢q^⁢(s1).

Proof

We denote

J1=∫0π[bu1(s1-x,x)+bp(s1+h)-bp(s1+h-2x)+εv1(s1-x,x)+εq(s1+h)-εq(s1+h-2x)+f(s1-x,x,u1(s1-x,x)+p(s1+h)-p(s1+h-2x))]dx,

J2=∫0π[bu1(s1+x,x)+bp(s1+h+2x)-bp(s1+h)+εv1(s1+x,x)+εq(s1+h+2x)-εq(s1+h)+f(s1+x,x,u1(s1+x,x)+p(s1+h+2x)-p(s1+h))]dx.

Since (u,v) is a solution of L⁢u+h1⁢(t,x)=0, and recalling the notation of the function h1⁢(t,x) occurs in (5.7), then we note that

∫0πh1⁢(s1+h-x,x)⁢dx=∫0πh1⁢(s1+h+x,x)⁢dx and ∫0πh1⁢(s1-x,x)⁢dx=∫0πh1⁢(s1+x,x)⁢dx

by taking into account (5.3) with t=s1+h and t=s1 respectively. Then we have

(6.6)J=(J1-∫0πh1⁢(s1+h-x,x)⁢dx)+(∫0πh1⁢(s1+h+x,x)⁢dx-J2)+(J2-∫0πh1⁢(s1+x,x)⁢d⁢x):=J~1+J~2+J~3.

Behavior of J~1 and J~2: We begin with the study of the asymptotic behavior of the first term in the right-hand side of (6.6). It is plain to check that

J~1=∫0π[bu1(s1-x,x)-bu1(s1+h-x,x)+εv1(s1-x,x)-εv1(s1+h-x,x)-f⁢(s1+h-x,x,u1⁢(s1+h-x,x)+p⁢(s1+h)-p⁢(s1+h-2⁢x))+f(s1-x,x,u1(s1-x,x)+p(s1+h)-p(s1+h-2x))]dx.

By virtue of (6.1) and f∈C1⁢(Ω×R), we get

(6.7)|J~1|≤∫0π[C⁢|h|+C⁢|u1⁢(s1-x,x)-u1⁢(s1+h-x,x)|]⁢dx≤C⁢|h|,

where C>0 is independent of ℎ. Moreover, a similar calculation shows that

(6.8)|J~2|≤∫0π|bu1(s1+h+x,x)-bu1(s1+x,x)+εv1(s1+h+x,x)-εv1(s1+x,x)+f⁢(s1+h+x,x,u1⁢(s1+h+x,x)+p⁢(s1+h+2⁢x)-p⁢(s1+h))-f(s1+x,x,u1(s1+x,x)+p(s1+h+2x)-p(s1+h))|dx≤C|h|.

Behavior of J~3: Using the notations of p^⁢(t)=p⁢(t+h)-p⁢(t) and q^⁢(t)=q⁢(t+h)-q⁢(t), we can represent

J~3=∫0π[bp^(s1+2x)-bp^(s1)+εq^(s1+2x)-εq^(s1)+f⁢(s1+x,x,u1⁢(s1+x,x)+p⁢(s1+h+2⁢x)-p⁢(s1+h))-f(s1+x,x,u1(s1+x,x)+p(s1+2x)-p(s1))]dx.

By (6.2) and the definition of M1, we obtain

p^⁢(s1+2⁢x)≤M1<p^⁢(s1)+M1⁢|h|,

which indicates that

(6.9)p⁢(s1+h+2⁢x)-p⁢(s1+h)<p⁢(s1+2⁢x)-p⁢(s1)+M1⁢|h|.

In addition, from the facts that 𝑞 is 2⁢π-periodic and ∫02⁢πq⁢(τ)⁢dτ=0, we deduce that

∫0πq^⁢(s1+2⁢x)⁢dx=∫0π[q⁢(s1+h+2⁢x)-q⁢(s1+2⁢x)]⁢dx=0.

Thus the above estimates give

(6.10)∫0π[b⁢p^⁢(s1+2⁢x)-b⁢p^⁢(s1)+ε⁢q^⁢(s1+2⁢x)-ε⁢q^⁢(s1)]⁢dx≤π⁢b⁢M1⁢|h|-ε⁢π⁢q^⁢(s1).

Moreover, we infer from (6.9), 4 and f∈C1⁢(Ω×R) that

(6.11)∫0π[f(s1+x,x,u1(s1+x,x)+p(s1+h+2x)-p(s1+h))-f(s1+x,x,u1(s1+x,x)+p(s1+2x)-p(s1))]dx≤CM1π|h|.

Therefore, making use of (6.10) and (6.11), we have

(6.12)J~3≤π⁢b⁢M1⁢|h|-ε⁢π⁢q^⁢(s1)+C⁢M1⁢π⁢|h|.

Now, inserting estimates (6.7), (6.8) and (6.12) into (6.6), we obtain (6.5). ∎

Step 2

We require the next lemma concerning the properties of the integrals

R1=∫0π[f(s1-x,x,u1(s1-x,x)+p(s1+h)-p(s1+h-2x))-f(s1-x,x,u1(s1-x,x)+p(s1)-p(s1-2x))]dx,

R2=∫0π[g(s2-x,x,v1(s2-x,x)+q(s2+h)-q(s2+h-2x))-g(s2-x,x,v1(s2-x,x)+q(s2)-q(s2-2x))]dx.

Lemma 6.2

Under the assumptions of Theorem 1.5, we have R1+R2≤C⁢|h| for 𝜀 is sufficiently small, where C>0 is independent of ℎ.

Proof

Noting that ∫0πp^⁢(s1-2⁢x)⁢dx=∫0πq^⁢(s1-2⁢x)⁢dx=0, then, by a direct computation, we observe that

R1=J-∫0π[b⁢p^⁢(s1)-b⁢p^⁢(s1-2⁢x)+ε⁢q^⁢(s1)-ε⁢q^⁢(s1-2⁢x)]⁢dx=J-b⁢π⁢p^⁢(s1)-ε⁢π⁢q^⁢(s1).

Hence, with the help of Lemma 6.1, we have

(6.13)R1≤-b⁢π⁢p^⁢(s1)-2⁢ε⁢π⁢q^⁢(s1)+C⁢|h|.

Similar to the derivation of inequalities (6.5) and (6.13), we are able to get

(6.14)R2≤-b⁢π⁢q^⁢(s2)-2⁢ε⁢π⁢p^⁢(s2)+C⁢|h|.

Selecting |ε|<b2, we deduce from (6.2) and the definitions of M1,M2 that

b⁢p^⁢(s1)+2⁢ε⁢p^⁢(s2)≥b⁢M1⁢(1-|h|)-2⁢|ε|⁢M1>-b⁢M1⁢|h|,

b⁢q^⁢(s2)+2⁢ε⁢q^⁢(s1)≥b⁢M2⁢(1-|h|)-2⁢|ε|⁢M2>-b⁢M2⁢|h|.

By virtue of the above two inequalities, then adding up (6.13) and (6.14) will yield that R1+R2≤C⁢|h| for |ε|<b2. That is what we desire. ∎

Step 3

Now, we are in a position to prove (6.3). Under the assumptions of Theorem 1.5, we know ϕ⁢(τ), ψ⁢(τ) are strictly increasing in 𝜏 and ϕ⁢(0)=0, ψ⁢(0)=0. Moreover, it follows from f,g∈C1 that ϕ⁢(τ), ψ⁢(τ) are Lipschitz continuous on any bounded intervals. By the definitions of R1, R2, 𝜙 and 𝜓, we conclude that

ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))≤f⁢(s1-x,x,u1⁢(s1-x,x)+p⁢(s1+h)-p⁢(s1-2⁢x+h))-f⁢(s1-x,x,u1⁢(s1-x,x)+p⁢(s1)-p⁢(s1-2⁢x)).

Then integrating the above in 𝑥 leads to

∫0πϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))⁢dx≤R1.

And a similar computation shows that ∫0πψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))⁢dx≤R2.

Therefore, we derive (6.3) from Lemma 6.2.∎

6.2.2 Proof of (6.4)

We denote by Σ1c, Σ2c the complement spaces of Σ1, Σ2 respectively. For any x∈[0,π], it follows from (6.2) and the choice of M1 that

p^⁢(s1)-p^⁢(s1-2⁢x)≥M1⁢(1-|h|)-M1=-M1⁢|h|.

Combining the above inequality with the facts that ϕ⁢(0)=0, 𝜙 is strictly increasing and Lipschitz continuous, we conclude that

ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))≥ϕ⁢(-M1⁢|h|)-ϕ⁢(0)≥-C⁢|h| for⁢x∈Σ1c.

Integrating the above in 𝑥 on Σ1c, we have

(6.15)-∫Σ1cϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))⁢dx≤C⁢|h|.

On the other hand, concerning the behavior of ψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x)) restricted on Σ2c, we also get

(6.16)-∫Σ2cψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))⁢dx≤C⁢|h|.

Hence the inequalities of (6.3), (6.15) and (6.16) ensure that

∫Σ1ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))⁢d⁢x+∫Σ2ψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))⁢d⁢x=(∫0π-∫Σ1c)ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x))⁢d⁢x+(∫0π-∫Σ2c)ψ⁢(q^⁢(s2)-q^⁢(s2-2⁢x))⁢d⁢x≤C⁢|h|,

where C>0 is independent of ℎ. Thus we arrive at (6.4).∎

6.3 Complete the Proof of Theorem 1.5

Finally, we examine the continuity of 𝑝 and 𝑞 by controlling the values of ϕ⁢(12⁢M1) and ψ⁢(12⁢M2).

Recalling the definitions of Σ1, Σ2 and the facts that ϕ⁢(τ), ψ⁢(τ) are strictly increasing in 𝜏, we deduce that ϕ⁢(12⁢M1)≤ϕ⁢(p^⁢(s1)-p^⁢(s1-2⁢x)) for x∈Σ1, and ψ⁢(12⁢M2)≤ϕ⁢(q^⁢(s2)-q^⁢(s2-2⁢x)) for x∈Σ2. Thus we infer from (6.4) that

(6.17)∫Σ1ϕ⁢(M12)⁢dx+∫Σ2ψ⁢(M22)⁢dx≤C⁢|h|.

We next claim that meas⁡Σ1>0 and meas⁡Σ2>0. Indeed, it follows from ∫02⁢πp^⁢(τ)⁢dτ=0 that

(6.18)0=∫0πp^⁢(s1-2⁢x)⁢dx=∫Σ1p^⁢(s1-2⁢x)⁢dx+∫Σ1cp^⁢(s1-2⁢x)⁢dx.

On the other hand, according to the definitions of M1, s1 and Σ1c, we deduce

(6.19)p^⁢(s1-2⁢x)≥{-M1for⁢x∈Σ1,p^⁢(s1)-M12>M12-M1⁢|h|for⁢x∈Σ1c.

Then, comparing with (6.18) and (6.19), we arrive at

0≥-M1⁢meas⁡Σ1+(12-|h|)⁢M1⁢(π-meas⁡Σ1)=-(32-|h|)⁢M1⁢meas⁡Σ1+(12-|h|)⁢M1⁢π,

which implies that

meas⁡Σ1>(1-2⁢|h|)⁢π3-2⁢|h|>π5 as⁢|h|<14.

A similar argument allows us to obtain meas⁡Σ2>π5, for |h|<14. Therefore, we make use of (6.17) and the preceding estimates for meas⁡Σ1, meas⁡Σ2 to get

ϕ⁢(M12)+ψ⁢(M22)≤C⁢|h|.

Furthermore, noting that ϕ⁢(12⁢M1)>0, ψ⁢(12⁢M2)>0 and ϕ⁢(τ), ψ⁢(τ) are strictly increasing in 𝜏, we conclude that M1→0 and M2→0. Hence it follows that y=p⁢(t+x)-p⁢(t-x)∈C⁢(Ω) and z=q⁢(t+x)-q⁢(t-x)∈C⁢(Ω). We finish the proof of Theorem 1.5.∎

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11701310

Award Identifier / Grant number: 11771428

Award Identifier / Grant number: 12031015

Award Identifier / Grant number: 12026217

Funding source: Natural Science Foundation of Shandong Province

Award Identifier / Grant number: ZR2016AQ04

Funding source: Qingdao Agricultural University

Award Identifier / Grant number: 6631114328

Funding statement: The research was supported by the National Natural Science Foundation of China (11701310, 11771428, 12031015, 12026217), the Natural Science Foundation of Shandong Province (ZR2016AQ04) and the Research Foundation for Advanced Talents of Qingdao Agricultural University (6631114328).

Acknowledgements

The authors are grateful to the referee for his/her useful comments.

Communicated by: Zhi-Qiang Wang

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Received: 2021-03-01

Revised: 2021-06-25

Accepted: 2021-06-25

Published Online: 2021-07-17

Published in Print: 2021-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2021-2138

Keywords for this article

Wave Equation; Variational Method; Klein–Gordon System; Periodic Solutions

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