Small amplitude periodic solutions in time for one-dimensional nonlinear wave equations

Abstract

This paper is devoted to the construction of solutions for one-dimensional wave equations with Dirichlet or Neumann boundary conditions by means of a Nash-Moser iteration scheme, for a large set of frequencies.

1 Introduction

We consider one-dimensional nonlinear wave equations like

$$\begin{aligned} u_{tt}-u_{xx}+mu=\varepsilon g(x,\omega t,u), \quad x\in [0,\pi ],\ t\in {\mathbb {R}}, \end{aligned}$$

(1)

where $g(x,\cdot ,u)$ is a time-periodic external forcing with period $2\pi$, g(x, t, u) $\in \mathrm {C}^\kappa ([0,\pi ]\times {\mathbb {R}}\times {\mathbb {R}};{\mathbb {R}})$ for some $\kappa$ large enough, and $g(x,t,0)=0$; the mass $m\in {\mathbb {R}}^+$; $\varepsilon >0$ is a small amplitude parameter; $\omega$ is a frequency parameter; and the displacement $u: [0,\pi ]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is the unknown. In the present paper we want to consider both Dirichlet boundary conditions

$$\begin{aligned} u(0,t)=u(\pi ,t)=0, \quad t\in {\mathbb {R}}, \end{aligned}$$

(2)

and Neumann boundary conditions

$$\begin{aligned} u_x(0,t)=u_x(\pi ,t)=0, \quad t\in {\mathbb {R}}. \end{aligned}$$

(3)

The existence of Cantor families of periodic solutions of the nonlinear wave equations have been studied by many authors, for example, see [2, 3, 7–13] and the references therein. Recently, Berti and Bolle [11] have proved the existence of Cantor families of spatial periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions of the form

$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt}-\triangle u+mu=\varepsilon F(\omega t,x,u),\\ u(t,x)=u(t,x+2\pi k), \quad \forall k\in \mathbb {Z}^d. \end{array}\right. \end{aligned}$$

Biasco and Gregorio [13] have studied the periodic in time solutions of the one-dimensional autonomous nonlinear wave equation with Dirichlet boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt}-u_{xx}+\mu u+f(u)=0,\\ u(t,0)=u(t,\pi )=0, \end{array}\right. \end{aligned}$$

where $\mu > 0$ is the mass and the nonlinearity f is an odd, real analytic function with $f'(0)=0,\ f'''(0)\ne 0$.

In addition, there are many other references, but most of them studied on the nonlinear wave equations with Dirichlet boundary conditions. The proofs of all above results rely on the use of the Nash-Moser implicit function theorem, to overcome unavoidable losses of derivatives coming from the small divisors appearing when inverting the linear part of the equation. In order to construct the existence of periodic and quasi-periodic solutions to nonlinear wave equations, this main difficulty, namely the presence of small divisors in the expansion series of the solutions, can be handled by KAM theory (see, e.g., [5, 6, 14, 20]), Lindstedt series method (see, e.g., [15–18]), and Nash-Moser iteration (see, e.g., [1–3, 7–11]).

The principle objective here is to look for small amplitude, $2\pi /\omega$-periodic in time solutions of Eq. (1) under Dirichlet boundary conditions (2) or Neumann boundary conditions (3) for all frequencies $\omega$ in some set of positive measure. The small divisors problem is overcome thanks to employing Nash-Moser iteration techniques.

The organization of the paper is described as follows. The next section states the main theorem on existence of Cantor families of time-periodic solutions of the system (1)–(2) or (1)–(3). In Sect. 3, we construct the solutions to the systems by making use of suitable Nash-Moser iteration scheme, and give the proof of theorem afterward in Sect. 4. The last section is devoted to showing the invertibility of linearized problem via the eigenvalues technique.

2 Statement of the main theorem

We denote by

$$\begin{aligned} \{\lambda _i | 0<\lambda _1 < \lambda _2\le \cdots \le \lambda _k \le \cdots \} \end{aligned}$$

and

$$\begin{aligned} \{\tilde{\lambda }_j | 0<\tilde{\lambda }_1 <\tilde{\lambda }_2 \le \cdots \le \tilde{\lambda }_k \le \cdots \} \end{aligned}$$

respectively, the eigenvalues of Dirichlet boundary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\varphi ''_i+m\varphi _i=\lambda _i^2\varphi _i, \quad x\in (0,\pi ),\\ \varphi _i(0)=\varphi _i(\pi )=0, \end{array}\right. \end{aligned}$$

(4)

and the eigenvalues of Neumann boundary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\tilde{\varphi }''_j+m\tilde{\varphi }_j=\tilde{\lambda }_j^2\tilde{\varphi }_j, \quad x\in (0,\pi ),\\ \tilde{\varphi }'_j(0)=\tilde{\varphi }'_j(\pi )=0, \end{array}\right. \end{aligned}$$

(5)

where $\varphi _i(x)$ and $\tilde{\varphi }_j(x)$ are the corresponding eigenfunctions, respectively.

Normalizing the period to $2\pi$, (1) can be written to

$$\begin{aligned} \omega ^2u_{tt}-u_{xx}+mu=\varepsilon g(x,t,u), \quad x\in [0,\pi ],\ t\in {\mathbb {R}}. \end{aligned}$$

(6)

We look for periodic solutions of (1)–(2) in the Banach spaces

$$\begin{aligned} X_{\sigma ,s}:= & {} \left\{ u(x,t)=\sum _{j=1}^\infty u_j(t)\varphi _j(x)\ \Bigg | \ u_j\in H^1({\mathbb {R}},{\mathbb {R}}),\right. \nonumber \\&\left. \Vert u\Vert _{\sigma ,s}^2=\sum _{j=1}^\infty \Vert u_j\Vert _{H^1}^2j^{2s}\mathrm {e}^{2\sigma j}<+\infty \right\} \end{aligned}$$

(7)

or look for periodic solutions of (1)–(3) in the Banach spaces as follows

$$\begin{aligned} \tilde{X}_{\sigma ,s}:= & {} \left\{ u(x,t)=\sum _{j=1}^\infty \tilde{u}_j(t)\tilde{\varphi }_j(x)\ \Bigg | \ \tilde{u}_j\in H^1({\mathbb {R}},{\mathbb {R}}),\right. \nonumber \\&\left. \Vert u\Vert _{\sigma ,s}^2=\sum _{j=1}^\infty \Vert \tilde{u}_j\Vert _{H^1}^2j^{2s}\mathrm {e}^{2\sigma j}<+\infty \right\} \end{aligned}$$

(8)

where $s>1/2,\ \sigma \ge 0$.

For convenience, the spaces $X_{\sigma ,s}$ and $\tilde{X}_{\sigma ,s}$, the eigenvalues $\lambda$ and $\tilde{\lambda }$, the eigenfunctions $\varphi$ and $\tilde{\varphi }$, are unified into $X_{\sigma ,s}$, $\lambda$, and $\varphi$, respectively. And we write $X_{\sigma ,s}$ for $X_s$, $\Vert u\Vert _{\sigma ,s}$ for $\Vert u\Vert _s$. For $s>1/2$, $X_s$ is a multiplicative Banach algebra (the proof is as in [4], Appendix 6.5), namely

$$\begin{aligned} \forall u_1, u_2\in X_s\Rightarrow u_1u_2\in X_s,\quad \mathrm{and}\quad \Vert u_1u_2\Vert _s\le C(s)\Vert u_1\Vert _s\Vert u_2\Vert _s. \end{aligned}$$

For the two boundary value problems above, we can get the same conclusion as follows.

Theorem 1

For the fixed $0<\bar{\omega }_1<\bar{\omega }_2$, there are $s,\kappa \in \mathbb {N}$, such that $\forall g\in \mathrm {C}^\kappa ([0,\pi ]\times {\mathbb {R}}\times {\mathbb {R}})$, $\forall \gamma \in (0,\lambda _1)$, there exist $\varepsilon _0, K, C>0$, a map $\tilde{u}\in \mathrm {C}^1\left( [0,\varepsilon _0]\times [\bar{\omega }_1,\bar{\omega }_2]; X_0\right)$ with

$$\begin{aligned} \Vert \tilde{u}\Vert _0\le \frac{\varepsilon K}{\gamma },\qquad \Vert \mathrm {D}_\varepsilon \tilde{u}\Vert _0\le \frac{K}{\gamma },\qquad \Vert \mathrm {D}_\omega \tilde{u}\Vert _0\le \frac{\varepsilon K}{\gamma ^2}, \end{aligned}$$

and a Cantor like set $A_\infty \subset [0,\varepsilon _0]\times [\bar{\omega }_1,\bar{\omega }_2]$ such that $\tilde{u}(\varepsilon ,\omega )$ is a solution of (1)–(2) or (1)–(3).

Moreover, the set $A_\infty$ satisfies Lebesgue measure property:

$$\begin{aligned} |A_\infty |\ge \varepsilon _0(\bar{\omega }_2-\bar{\omega }_1)(1-C\gamma ), \end{aligned}$$

i.e. $\lim _{\gamma \rightarrow 0}\left( |A_\infty |/\varepsilon _0(\bar{\omega }_2-\bar{\omega }_1)\right) =1$.

3 The Nash-Moser iteration scheme

Consider the orthogonal splitting $X_s=X^{(n)}\oplus X^{(n)\bot }$, where

$$\begin{aligned} X^{(n)}:= & {} \Bigg \{u\in X_s \Bigg | u(x,t)=\sum _{\lambda _j\le N_n} u_j(t)\varphi _j(x)\Bigg \},\\ X^{(n)\perp }:= & {} \Bigg \{u\in X_s \Bigg | u(x,t)=\sum _{\lambda _j> N_n} u_j(t)\varphi _j(x)\Bigg \}, \end{aligned}$$

with $N_n:=N_02^n,\ n\in \mathbb {N}$, and $N_0\in \mathbb {N}$ large enough.

The convergence of the Nash-Moser scheme is based on properties (P1)–(P5) below. The first three properties are standard for the composition operator $f: X_s\rightarrow X_s$ defined by $f(u)(x,t):=g(x,t,u(x,t))$.

• (P1) (Regularity) $f \in \mathrm {C}^2(X_s ; X_s )$ and $f,\ \mathrm {D}f$, $\mathrm {D}^2 f$ are bounded on $\{\Vert u\Vert _s \le 1\}$.

• (P2) (Tame) $\forall s\le s'\le k$, $\forall u\in X_{s'}$ such that $\Vert u\Vert _s\le 1$, $\Vert f(u)\Vert _{s'}\le C(s')(1+\Vert u\Vert _{s'})$.

• (P3) (Taylor Tame) $\forall s\le s'\le k-2$, $\forall u, h\in X_{s'}$ such that $\Vert u\Vert _s, \Vert h\Vert _s \le 1$,

  $$\begin{aligned} \left\| f(u+h)-f(u)-\mathrm {D}f(u)[h]\right\| _{s'}\le C(s')\left( \Vert u\Vert _{s'}\Vert h\Vert _s^2+\Vert h\Vert _s\Vert h\Vert _{s'}\right) , \end{aligned}$$

  Where $\mathrm {D}f(u)=\partial _ug(x,t,u(x,t))$. In particular, for $s'=s$,

  $$\begin{aligned} \left\| f(u+h)-f(u)-\mathrm {D}f(u)[h]\right\| _s\le C\Vert h\Vert _s^2. \end{aligned}$$

  We refer to the references [19, 21] for the proof of (P2). Properties (P1) and (P3) are obtained similarly.

  We assume the existence of orthogonal projectors respectively onto $X^{(n)}$ and $X^{(n)\bot }$ denoted by $P_n: X\rightarrow X^{(n)}$ and $P_n^\perp : X\rightarrow X^{(n)\bot }$.

• (P4) (Smoothing) For all $s,r>0,\ \forall n\in \mathbb {N}$, there hold

  $$\begin{aligned} \Vert P_nu\Vert _{s+r}\le N_n^r\Vert u\Vert _s, \quad \forall u\in X_s, \end{aligned}$$

  (9)
  $$\begin{aligned} \Vert (I-P_n) u\Vert _s\le N_n^{-r}\Vert u\Vert _{s+r}, \quad \forall u\in X_{s+r}, \end{aligned}$$

  (10)

  where I is the identity map.

  The key property (P5), proved in Sect. 5, is an invertibility property for the linearized operator

  $$\begin{aligned} {\mathcal {L}}_n\left( \varepsilon ,\omega ,u(\varepsilon ,\omega )\right) [h]:=L_\omega h-\varepsilon P_n\mathrm {D}f(u(\varepsilon ,\omega ))h, \end{aligned}$$

  (11)

  where $L_\omega :=\omega ^2\partial _t^2-\partial _x^2+m$.

  Denote ${\mathfrak {J}}_n:=\{j\in \mathbb {N}\big |\ 1<\lambda _j\le N_n\},\ n=0,1,2,\ldots .$

  For $\gamma \in (0,\lambda _1),\ \tau \ge 3$, we define

  $$\begin{aligned} A_0:= & {} \left\{ (\varepsilon ,\omega )\in [0,\varepsilon _0]\times [\bar{\omega }_1,\bar{\omega }_2]: |\omega p_l-\lambda _j|>\frac{\gamma }{j^\tau },\ \forall j\in {\mathfrak {J}}_0,\ l\in \mathbb {N}\right\} ,\\ A_{n+1}:= & {} \left\{ (\varepsilon ,\omega )\in A_n: |\omega p_l-\lambda _j|>\frac{\gamma }{j^\tau },\ \forall j\in {\mathfrak {J}}_{n+1},\ l\in \mathbb {N}\right\} \end{aligned}$$

  for every $n=0,1,2,\ldots$, where $p_l^2\ (l\in \mathbb {N})$ are eigenvalues of the following problem

  $$\begin{aligned} \left\{ \begin{array}{ll} y''+p^2y=0, \\ y(t)=y(t+\pi ). \end{array}\right. \end{aligned}$$

  (12)

• (P5) (Invertibility of ${\mathcal {L}}_n$) Assume that $u\in X^{(n)}$, $(\varepsilon ,\omega )\in A_{n+1}$, there exist positive constants $\delta _0, K'$ such that $\varepsilon /\gamma <\delta _0$, then ${\mathcal {L}}_n$ is invertible and

  $$\begin{aligned} \Vert {\mathcal {L}}_n^{-1}(\varepsilon ,\omega ,u)h\Vert _0\le \frac{K'}{\gamma }N_n^{\tau }\Vert h\Vert _0, \qquad \forall h\in X^{(n+1)}. \end{aligned}$$

  (13)

Proof

The proof will be given in Sect. 5. $\square$

Lemma 1

(initialization) For $(\varepsilon ,\omega )\in A_0$, there are positive constants $K_0, \delta _1$ such that $\varepsilon /\gamma <\delta _1$, then there exists a solution $u_0:=u_0(\varepsilon ,\omega )\in X^{(0)}$ of equation $L_\omega u=\varepsilon P_0f(u)$ satisfying $\Vert u_0\Vert _0\le \varepsilon K_0/\gamma$.

Proof

Since the eigenvalues of $L_\omega$ satisfy

$$\begin{aligned} |\omega p_l-\lambda _j|>\frac{\gamma }{j^\tau },\ \forall j\in {\mathfrak {J}}_0,\ l\in \mathbb {N}, \end{aligned}$$

so $L_\omega$ is invertible on $X^{(0)}$ and, for some $K_1$,

$$\begin{aligned} \Vert L_\omega ^{-1}h\Vert _0\le \frac{K_1N_0^\tau }{\gamma }\Vert h\Vert _0, \qquad \forall h\in X^{(0)}. \end{aligned}$$

By the contraction mapping theorem, using the property (P1), for $\varepsilon /\gamma$ small, there exists a unique solution $u_0 := u_0(\varepsilon ,\omega )$ of equation $L_\omega u=\varepsilon P_0f(u)$ satisfying $\Vert u_0\Vert _0\le \varepsilon K_0/\gamma$. $\square$

For $(\varepsilon ,\omega )\in A_n, n\ge 1$, we construct a sequence $\{u_n\}_{n=0}^\infty$ by

$$\begin{aligned} u_{n+1}=u_n-{\mathcal {L}}_{n+1}(u_n)^{-1}\left[ L_\omega u_n-\varepsilon P_{n+1}f(u_n)\right] , \end{aligned}$$

(14)

and let $h_0=u_0,\ h_{n+1}=u_{n+1}-u_n,\ n=0,1,2,\ldots$.

Lemma 2

(induction step)  There exist $K_2, \beta :=4\tau -2$, and $\delta _2$ small enough. Assume that $h_k\in X^{(k)}$ for all $k=1,2,\ldots ,n$ satisfy

$$\begin{aligned} \Vert h_k\Vert _0<\frac{\varepsilon K_2}{\gamma }N_{k-1}^{-\beta }; \end{aligned}$$

and $u_n$ defined in (14) solve $L_\omega u=\varepsilon P_nf(u)$ for all $n=0,1,2,\ldots$. If $(\varepsilon ,\omega )\in A_{n+1}$ and $\varepsilon /\gamma <\delta _2$, then there exists $h_{n+1}\in X^{(n+1)}$ satisfying

$$\begin{aligned} \Vert h_{n+1}\Vert _0<\frac{\varepsilon K_2}{\gamma }N_n^{-\beta }. \end{aligned}$$

(15)

Proof

Taking into account $L_\omega u_n=\varepsilon P_nf(u_n)$, for $h_{n+1}$, we have

$$\begin{aligned} {\mathcal {L}}_{n+1}(u_n)h_{n+1}=\varepsilon (P_{n+1}-P_n)f(u_n)+\varepsilon P_{n+1}Q(u_n,h_{n+1}), \end{aligned}$$

(16)

where

$$\begin{aligned} Q(u_n,h_{n+1})=f(u_n+h_{n+1})-f(u_n)-\mathrm {D}f(u_n)h_{n+1}. \end{aligned}$$

Consider the fixed point problem

$$\begin{aligned} h_{n+1}=\varepsilon {\mathcal {L}}_{n+1}(u_n)^{-1}[(P_{n+1}-P_n)f(u_n)+P_{n+1}Q]:=\mathcal {G}(h_{n+1}) \end{aligned}$$

for $h_{n+1}\in X^{(n+1)}$. We shall prove that $\mathcal {G}$ is a contraction. By (14) and the properties (P1)–(P4),

$$\begin{aligned} \Vert \mathcal {G}(h_{n+1})\Vert _0&\le \frac{\varepsilon K'}{\gamma }\left( \Vert (P_{n+1}-P_n)f(u_n)\Vert _{\tau -1}+\Vert P_{n+1}Q\Vert _{\tau -1}\right) \\&\le \frac{\varepsilon K'}{\gamma }\left( N_n^{-\beta }\Vert P_{n+1}f(u_n)\Vert _{\tau -1+\beta }+C_1\Vert h_{n+1}\Vert _{\tau -1}^2\right) \\&\le \frac{\varepsilon K'}{\gamma }\left( N_n^{-\beta }C_2(1+\Vert u_n\Vert _{\tau -1+\beta })+C_1\Vert h_{n+1}\Vert _{\tau -1}^2\right) \\&\le \frac{\varepsilon K'}{\gamma }\left( N_n^{-\beta }C_3+C_1N_{n+1}^{2(\tau -1)}\Vert h_{n+1}\Vert _0^2\right) . \end{aligned}$$

If $\Vert h_{n+1}\Vert _0<\rho _{n+1}:=(\varepsilon K_2/\gamma )N_n^{-\beta }$, then $\Vert \mathcal {G}(h_{n+1})\Vert _0 \le \rho _{n+1}$ for $\varepsilon /\gamma$ small enough, i.e. $\mathcal {G}(B_{n+1})\subseteq B_{n+1}:=\{\Vert h\Vert _0<\rho _{n+1}\}$. Therefore the lemma follows from the contraction mapping theorem. $\square$

4 Proof of Theorem

The goal of this section is to prove our main result based on Sect. 3.

Lemma 3

(existence of solution)  Suppose that $A_\infty :=\bigcap _{n\ge 0} A_n \ne \varnothing$. If $(\varepsilon ,\omega )\in A_\infty$ and $\varepsilon /\gamma <\delta _3$ small enough, then the sequence $\{u_n\}_{n=0}^\infty$ converges in $X_{0}$ to $u_\infty :=\sum _{n\ge 0}h_n$. $u_\infty$ is a solution of the equation (6) and

$$\begin{aligned} \Vert u_\infty \Vert _0\le \frac{\varepsilon K}{\gamma } \end{aligned}$$

(17)

for some K.

Proof

By Lemmas 1 and 2, the series $\sum _{n\ge 0}h_n$ converges, $u_n$ converges to $u_\infty$ in $X_0$ and (17) holds true.$\square$

Lemma 4

Assume the hypotheses of Lemma 2, then there exists constant $K_3$ such that

$$\begin{aligned} \Vert \partial _t^2h_0\Vert _0\le \frac{\varepsilon K_3}{\gamma \omega ^2}, \qquad \Vert \partial _t^2h_{n+1}\Vert _0\le \frac{\varepsilon K'_3}{\gamma \omega ^2}N_{n+1}^{-\beta },\quad n=0,1,2,\ldots . \end{aligned}$$

Proof

Note that $h_0=u_0$ solves $L_\omega u=\varepsilon P_0f(u)$. It implies

$$\begin{aligned} \omega ^2 \partial _t^2 h_0 +\lambda _jh_0=\varepsilon P_0f(u). \end{aligned}$$

Thus $\Vert \partial _t^2 h_0\Vert _0\le \varepsilon K_3/\gamma \omega ^2$ for some $K_3$.

It follows from (11) and (14) that

$$\begin{aligned} \omega ^2 \partial _t^2 h_{n+1}=\varepsilon (P_{n+1}-P_n)f(u_n)+\partial _x^2h_{n+1}-m h_{n+1}-\varepsilon P_{n+1}\mathrm {D}f(u_n)h_{n+1}. \end{aligned}$$

Hence, by (P2), (9) and (15), there exists some constant $K'_3$ such that

$$\begin{aligned} \omega ^2\Vert \partial _t^2 h_{n+1}\Vert _0&\le \varepsilon N_n^{-\beta }\Vert P_{n+1}f(u_n)\Vert _\beta +N_{n+1}^2\Vert h_{n+1}\Vert _0+\varepsilon \Vert \mathrm {D}f(u_n)h_{n+1}\Vert _0\\&\le \frac{\varepsilon K'_3}{\gamma }N_{n+1}^{-\beta } \end{aligned}$$

for $n=0,1,2,\ldots$. It completes the proof. $\square$

Lemma 5

(estimate of the derivatives)  Assume the hypotheses of Lemma 3, then $\mathrm {D}_{\varepsilon ,\omega }u_n$ converges to $\mathrm {D}_{\varepsilon ,\omega }u_\infty$ in $X_{0}$ satisfying

$$\begin{aligned} \Vert \mathrm {D}_\varepsilon u_\infty \Vert _0\le \frac{K}{\gamma },\qquad \Vert \mathrm {D}_\omega u_\infty \Vert _0\le \frac{\varepsilon K}{\gamma ^2}. \end{aligned}$$

(18)

Proof

By the proof of Lemma 1 and the implicit function theorem, $(\varepsilon ,\omega )\mapsto u_0(\varepsilon ,\omega )$ is in $\mathrm {C}^1(A_0, X_0)$ and $\Vert \mathrm {D}_{\varepsilon ,\omega }u_0\Vert _0\le K_0/\gamma$.

Next, assume by induction that $h_n(\varepsilon , \omega )$ is a $\mathrm {C}^1$ map defined in $A_n$ for every $n=0,1,2,\ldots$. We shall prove that $h_{n+1}(\varepsilon , \omega )$ is $\mathrm {C}^1$ too. Recall that $h_{n+1}$ is defined for $(\varepsilon , \omega )\in A_{n+1}$ as a solution in $X^{(n+1)}$ of Eq. (16). We claim that the operator

$$\begin{aligned} {\mathcal {L}}_{n+1}(u_{n+1})[z]:=L_\omega z-\varepsilon P_{n+1}\mathrm {D}f(u_n+h_{n+1}))[z] \end{aligned}$$

(19)

is invertible. In fact,

$$\begin{aligned} \Vert \left[ {\mathcal {L}}_{n+1}(u_{n+1})-{\mathcal {L}}_{n+1}(u_n)\right] h_{n+1}\Vert _0\le \varepsilon \Vert f(u_n)h_{n+1}\Vert _0\le \frac{\varepsilon ^2 K_2'}{\gamma }N_n^{-\beta }, \end{aligned}$$

which together with (13) gives

$$\begin{aligned} \Vert {\mathcal {L}}_{n+1}(u_{n})^{-1}\left[ {\mathcal {L}}_{n+1}(u_{n+1})-{\mathcal {L}}_{n+1}(u_n)\right] \Vert _0\le \frac{\varepsilon ^2 K'K_2'}{\gamma ^2}N_n^{\tau -\beta }. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert {\mathcal {L}}_{n+1}(u_{n+1})^{-1}\left[ {\mathcal {L}}_{n+1}(u_{n+1})-{\mathcal {L}}_{n+1}(u_n)\right] \Vert _0\le \frac{1}{2} \end{aligned}$$

provided that $\varepsilon /\gamma$ is appropriate small, while n is appropriate large enough. This shows that ${\mathcal {L}}_{n+1}(w_{n+1})$ is invertible and

$$\begin{aligned} \Vert {\mathcal {L}}_{n+1}(u_{n+1})^{-1}\Vert _0\le \frac{2K'}{\gamma }N_{n+1}^{\tau }. \end{aligned}$$

As a consequence, by the implicit function theorem, the map $(\varepsilon ,\omega )\mapsto h_{n+1}(\varepsilon ,\omega )$ is in $\mathrm {C}^1(A_{n+1}, X^{(n+1)})$.

By (16),

$$\begin{aligned} L_\omega h_{n+1}-\varepsilon P_{n+1}[f(u_n+h_{n+1})-f(u_n)]-\varepsilon (P_{n+1}-P_n)f(u_n)=0. \end{aligned}$$

(20)

Differentiating the Eq. (20) with respect to $\omega$ and utilizing ${\mathcal {L}}_{n+1}(u_{n+1})^{-1}$, then taking the norm $\Vert \cdot \Vert _0$ on both sides, we obtain

$$\begin{aligned} \big \Vert \partial _\omega h_{n+1}\big \Vert _0\le & {} \frac{2K'}{\gamma }N_{n+1}^{\tau }\Big (\Vert 2\omega (\partial _t^2h_{n+1})\Vert _0 +\Vert \varepsilon (P_{n+1}-P_n)\mathrm {D}f(u_n)\partial _\omega u_n\Vert _0\\&+\Vert \varepsilon P_{n+1}\left[ \mathrm {D}f(u_n+h_{n+1})-\mathrm {D}f(u_n)\right] \partial _\omega u_n\Vert _0\Big )\\\le & {} \frac{2\varepsilon K'}{\gamma }N_{n+1}^{\tau }\Big (\frac{2K'_3}{\gamma \omega }N_{n+1}^{-\beta } + C_4N_n^{-\beta }\sum _{i=0}^n\Vert \partial _\omega h_i\Vert _0 +C_5\rho _{n+1}\sum _{i=0}^n\Vert \partial _\omega h_i\Vert _0 \Big )\\\le & {} \frac{\varepsilon K}{\gamma ^2}N_{n+1}^{\tau }N_n^{-\beta }. \end{aligned}$$

Hence, we deduce $\Vert \partial _\omega u_{n+1}\Vert _0\le \varepsilon K/\gamma ^2$ which implies $\Vert \mathrm {D}_\omega u_\infty \Vert _0\le \varepsilon K/\gamma ^2$.

Similarly, differentiating the Eq. (20) with respect to $\varepsilon$ gives

$$\begin{aligned} {\mathcal {L}}_{n+1}(u_{n+1})[\partial _\varepsilon h_{n+1}]&= P_nf(u_n)-P_{n+1}f(u_{n+1})\\&\quad \ +\,\varepsilon P_n\mathrm {D}f(u_n)\partial _\varepsilon u_n-\varepsilon P_{n+1}\mathrm {D}f(u_{n+1})\partial _\varepsilon u_n, \end{aligned}$$

and then, we can obtain the estimate for $\partial _\varepsilon h_{n+1}$ by using the same method as above. $\square$

Finally we can define, by means of a cut-off function, a $\mathrm {C}^1$-Whitney extension $\tilde{u}_{n+1}\in \mathrm {C}^1(A_0,X^{(n+1)})$ of $u_{n+1}$ as $\tilde{u}_{n+1}:=\tilde{u}_n+\tilde{h}_{n+1}$ satisfying

$$\begin{aligned} \Vert \tilde{u}\Vert _0\le \frac{\varepsilon K}{\gamma },\qquad \Vert \mathrm {D}_\varepsilon \tilde{u}\Vert _0\le \frac{K}{\gamma },\qquad \Vert \mathrm {D}_\omega \tilde{u}\Vert _0\le \frac{\varepsilon K}{\gamma ^2}, \end{aligned}$$

where $\tilde{u}_n, \tilde{h}_{n+1}$ are obtained through the corresponding Whitney extension procedures.

Lemma 6

(measure estimate)  For $\tau \ge 3$, there exists $\delta <\min \{\delta _i, i=0,1,2,3\}$ such that the Cantor set $A_\infty$ has measure property: for every interval $(\bar{\omega }_1,\bar{\omega }_2)$ with $0<\bar{\omega }_1<\bar{\omega }_2<+\infty$, there is a constant C depending on $(\bar{\omega }_1,\bar{\omega }_2)$ such that $|A_\infty |\ge \varepsilon _0(\bar{\omega }_2-\bar{\omega }_1)(1-C\gamma ).$

Proof

Given $\varepsilon$, we need to prove that the complementary set $E:=\bigcup _{l,j\ge 1}\Omega _{l,j}$ has small measure, where

$$\begin{aligned} \Omega _{l,j}:=\left\{ \omega \in (\bar{\omega }_1,\bar{\omega }_2):\ |\omega p_l-\lambda _j|\le \frac{\gamma }{j^\tau } \right\} , \end{aligned}$$

and $\Omega _{0,j}=\varnothing$ for all $j\ge 1$.

Note that $l/4<\partial _\omega (\omega p_l)<2\gamma /j^\tau$ provided that $\varepsilon /\omega <\delta$ small enough. In addition,

$$\begin{aligned} \bar{\omega }_1l-\frac{\gamma }{j^\tau }<\lambda _j<\bar{\omega }_2l+\frac{\gamma }{j^\tau }, \end{aligned}$$

which implies

$$\begin{aligned} \sharp \{j\}=\frac{1}{\varrho }\left( (\bar{\omega }_2-\bar{\omega }_1)l+\frac{2\gamma }{j^\tau }\right) +1<Kl(\bar{\omega }_2-\bar{\omega }_1), \end{aligned}$$

where $\varrho :=\inf \{|\lambda _{j+1}-\lambda _j|:\ j\ge 1\}$.

For fixed $0<\bar{\omega }_1<\bar{\omega }_2<+\infty$, if $\Omega _{l,j}\cap (\bar{\omega }_1,\bar{\omega }_2)$ is nonempty, then

$$\begin{aligned} |E|\le \sum _{j=1}^\infty \frac{8\gamma }{lj^\tau }Kl(\bar{\omega }_2-\bar{\omega }_1)\le C\gamma (\bar{\omega }_2-\bar{\omega }_1) \end{aligned}$$

because the series $\sum _{j=1}^\infty 1/j^\tau$ converges. Thus

$$\begin{aligned} |A_\infty (\varepsilon )\cap (\bar{\omega }_1,\bar{\omega }_2)|\ge (\bar{\omega }_2-\bar{\omega }_1)(1-C\gamma ). \end{aligned}$$

Therefore

$$\begin{aligned} |A_\infty |\ge \int _0^{\varepsilon _0}(\bar{\omega }_2-\bar{\omega }_1)(1-C\gamma )\mathrm {d}\varepsilon =\varepsilon _0(\bar{\omega }_2-\bar{\omega }_1)(1-C\gamma ), \end{aligned}$$

and we get the thesis. $\square$

5 Inversion of the linearized operator

In this section, we prove the key property on the inversion of the linearized operator defined in (11). We also write the operator

$$\begin{aligned} {\mathcal {L}}_n=\mathcal {D}+\varepsilon \mathcal {M} \end{aligned}$$

with

$$\begin{aligned} \mathcal {D}h:= & {} \omega ^2h_{tt}-h_{xx}+mh,\\ \mathcal {M}h:= & {} P_n(ah),\quad a(x,t):=-\partial _ug(x,t,u(x,t)). \end{aligned}$$

Next, it is easy to show the result below.

Lemma 7

Let $p_l^2$ and $\psi _l\ (l\in \mathbb {N})$ be the eigenvalues and eigenfunctions of the problem (12), then the eigenfunctions $\psi _l$ form an orthonormal basis of $H^1([0,\pi ])$ with respect to the product $(u,v)_{H^1}=\int _0^{\pi }[u'v'+uv]\mathrm {d}t$.

Lemma 8

(inversion of $\mathcal {D}$)  Let $p_l^2(l\in \mathbb {N})$ be the eigenvalues of the problem (12). For all $j\in {\mathfrak {J}}_{n+1}$, If $(\varepsilon ,\omega )$ satisfies the conditions

$$\begin{aligned} \left| \omega p_l(\varepsilon ,\omega )-\lambda _j\right| >\frac{\gamma }{j^\tau },\qquad l\in \mathbb {N}, \end{aligned}$$

(21)

then $\mathcal {D}$ is invertible, and

$$\begin{aligned} \Vert \mathcal {D}^{-1}h\Vert _0 \le \frac{C}{\gamma }\Vert h\Vert _{\tau }, \quad \forall h\in X^{(n+1)} \end{aligned}$$

(22)

for some positive constant C.

Proof

We develop $\mathcal {D}h=\sum D_jh_j(t)\varphi _j(x),$ where

$$\begin{aligned} D_jz=\omega ^2z''+\lambda _j^2z=\sum _{l\in \mathbb {N}}\left( \lambda _j^2-\omega ^2p_l^2\right) \hat{z}_l\psi _l(t), \quad z=\sum _{l\in \mathbb {N}}\hat{z}_l\psi _l(t). \end{aligned}$$

Thus, each $D_j$ is the diagonal with respect to the basis $\psi _l(t)$. By (21), $\forall j\in {\mathfrak {J}}_{n+1}$, we have $D_j$ is invertible and

$$\begin{aligned} \left\| D_j^{-1}z\right\| _{H^1}= \sum _{l\in \mathbb {N}}\frac{1}{|\lambda _j^2-\omega ^2p_l^2|}\Vert z\Vert _{H^1} \le \frac{Cj^{\tau }}{\gamma }\Vert z\Vert _{H^1}, \end{aligned}$$

so that

$$\begin{aligned} \Vert \mathcal {D}^{-1}h\Vert _0^2\le \frac{C^2}{\gamma ^2}\Vert h\Vert _{\tau }^2 \end{aligned}$$

for some positive constant C. It implies (22) holds true. $\square$

Lemma 9

Assume the hypotheses of Lemma 8. Define $|\mathcal {D}|^{-1/2}: X^{(n)}\rightarrow X^{(n)}$ obeys

$$\begin{aligned} |\mathcal {D}|^{-1/2}h:=\sum _{1\le \lambda _j\le N_n}|D_j|^{-1/2}h_j(t)\varphi _j(x), \end{aligned}$$

then

$$\begin{aligned} \Big \Vert |\mathcal {D}|^{-1/2}h\Big \Vert _{\sigma ,s}\le \frac{K_4}{\sqrt{\gamma }}\Vert h\Vert _{\sigma ,s+\frac{\tau }{2}}\le \frac{K_4}{\sqrt{\gamma }}N_n^{\frac{\tau }{2}}\Vert h\Vert _{\sigma ,s}, \quad \forall h\in X^{(n)}. \end{aligned}$$

(23)

Proof

Due to $\Big \Vert |D_j|^{-1/2}z\Big \Vert _{H^1}\le (K'_4/\sqrt{\alpha _j})\Vert z\Vert _{H^1}$ for some $K'_4$, where

$$\begin{aligned} \alpha _j:=\min \limits _{l\in \mathbb {N}}\{|\lambda _j^2-\omega ^2p_l^2|\}>0, \end{aligned}$$

we have

$$\begin{aligned} \Big \Vert |\mathcal {D}|^{-1/2}h\Big \Vert _{\sigma ,s}^2&\le \sum _{1\le \lambda _j\le N_n}\frac{{K'_4}^2}{\alpha _j}\Vert h_j\Vert _{H^1}^2j^{2s}\mathrm {e}^{2\sigma j}\\&\le \sum _{1\le \lambda _j\le N_n}\frac{{K'_4}^2j^{\tau }}{\gamma }\Vert h_j\Vert _{H^1}^2j^{2s}\mathrm {e}^{2\sigma j}\\&\le \frac{{K'_4}^2}{\gamma }\Vert h\Vert ^2_{\sigma ,s+\frac{\tau }{2}}, \quad \forall h\in X^{(n)} \end{aligned}$$

whence (23) follows. In particular,

$$\begin{aligned} \Big \Vert |\mathcal {D}|^{-1/2}h\Big \Vert _0\le \frac{K_4}{\sqrt{\gamma }}\Vert h\Vert _{\frac{\tau }{2}}\le \frac{K_4}{\sqrt{\gamma }}N_n^{\frac{\tau }{2}}\Vert h\Vert _0. \end{aligned}$$

$\square$

6 Proof of (P5)

Proof

Let ${\mathcal {L}}_n(u)=|D|^{1/2}(\mathcal {U}+\varepsilon \mathcal {R})|D|^{1/2}$, where

$$\begin{aligned} \mathcal {U}:=|D|^{-1/2}\mathcal {D}|D|^{-1/2},\qquad \mathcal {R}:=|D|^{-1/2}\mathcal {M}|D|^{-1/2}. \end{aligned}$$

It is easily to prove that $\Vert \mathcal {U}\Vert _\sigma :=\sup _{\Vert h\Vert _\sigma \le 1}\Vert \mathcal {U}h\Vert _\sigma =1$.

Noting that

$$\begin{aligned} \alpha _j=\min \limits _{l\in \mathbb {N}}\left\{ |\omega ^2p_l^2-\lambda _j^2|\right\} \ge \min \limits _{l\in \mathbb {N}}\big \{\lambda _j|\omega p_l-\lambda _j|\big \}\ge \frac{\gamma }{j^{\tau }} \end{aligned}$$

for all $j\ge 1$, we get $\alpha _k\alpha _l\ge \gamma ^2/(kl)^{\tau },\ k,l\ge 1$.

For $h\in X^{(n)}$, $\mathcal {R}h=\sum _{\lambda _k\le N_n}(\mathcal {R}h)_k\mathrm {e}^{\mathrm {i}kt}$ with

$$\begin{aligned} (\mathcal {R}h)_k=|\mathcal {D}_k|^{-1/2}(\mathcal {M}|\mathcal {D}|^{-1/2}h)_k=|\mathcal {D}_k|^{-1/2}\big (P_na\sum _{\lambda _l\le N_n}|\mathcal {D}_l|^{-1/2}h_l\big )_k, \end{aligned}$$

we deduce

$$\begin{aligned} \Vert (\mathcal {R}h)_k\Vert _{H^1}\le C\sum _{\lambda _l\le N_n}\frac{\Vert a\Vert _{H^1}}{\sqrt{\alpha _k\alpha _l}}\Vert h_l\Vert _{H^1}\le \frac{C}{\gamma }S_k, \end{aligned}$$

where

$$\begin{aligned} S_k:=\sum _{\lambda _l\le N_n}\Vert a\Vert _{H^1}\bigg (\max \{k,l\}\bigg )^{\tau }\Vert h_l\Vert _{H^1}. \end{aligned}$$

Let $S(t):=\sum _{\lambda _k\le N_n}S_k\mathrm {e}^{\mathrm {i}kt}$, then

$$\begin{aligned} \Vert \mathcal {R}h\Vert _s^2=\sum _{\lambda _k\le N_n}\Vert (\mathcal {R}h)_k\Vert ^2_{H^1}k^{2s}\mathrm {e}^{2\sigma k} \le \frac{C^2}{\gamma ^2}\sum _{\lambda _k\le N_n}S_k^2 k^{2s}\mathrm {e}^{2\sigma k}=\frac{C^2}{\gamma ^2}\Vert S\Vert _s^2. \end{aligned}$$

It turns out that $S=P_n(bc)$ with

$$\begin{aligned} b(t):=\sum _{l\in \mathbb {N}}\Vert a\Vert _{H^1}\bigg (\max \{ k, l\}\bigg )^{\tau }\mathrm {e}^{\mathrm {i}lt} \end{aligned}$$

and

$$\begin{aligned} c(t):=\sum _{\lambda _l\le N_n}\Vert h_l\Vert _{H^1}\mathrm {e}^{\mathrm {i}lt}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \varepsilon \mathcal {R}h\Vert _s \le \frac{\varepsilon C}{\gamma }\Vert b\Vert _s\Vert c\Vert _s\le \frac{\varepsilon C}{\gamma }\Vert a\Vert _{s+\tau }\Vert h\Vert _s\le \frac{\varepsilon C'}{\gamma }\Vert h\Vert _s\le \frac{1}{2}\big \Vert h\big \Vert _s \end{aligned}$$

provided that we take $\varepsilon /\gamma$ small enough. Then Neumann series $\mathcal {U}+\varepsilon \mathcal {R}$ is invertible in $(X^{(n+1)}, \Vert \cdot \Vert _s)$, and $\Vert (\mathcal {U}+\varepsilon \mathcal {R})^{-1}h\Vert _s<2\Vert h\Vert _s$. Therefore

$$\begin{aligned} \Vert {\mathcal {L}}_n(u)^{-1}h\Vert _0=\bigg \Vert |D|^{-1/2}(\mathcal {U}+\varepsilon \mathcal {R})^{-1}|D|^{-1/2}h\bigg \Vert _0\le \frac{K'}{\gamma }N_n^{\tau }\Vert h\Vert _0. \end{aligned}$$

$\square$