Quasi-periodically forced and reversible vibrations of beam equations with Liouvillean frequencies

Abstract

The present paper is concerned with the existence of response solutions of quasi-periodic type for a class of quasi-periodically forced, non-Hamiltonian but reversible nonlinear beam equations. We do not suppose the basic frequency $\omega \in {\mathbb {R}}^2$ of the forcing term is Diophantine or Brjuno, and it might be Liouvillean, which is weaker than the Diophantine or Brjuno frequency. The proof is based on an improved Kolmogorov–Arnold–Moser (KAM) theorem for infinite-dimensional reversible systems with non-reducible normal form.

Appendix. Proofs of some technical lemmas

1.1 Proof of Lemma 5.2

Proof

• (1). In fact, by Lemma 2.1, one has $Q_{{\mathfrak {n}}+1}\ge Q_{{\mathfrak {n}}}^{{\mathcal {A}}}.$ ${\tilde{r}}_{m}\le {\tilde{r}}_{0}=2r_{+},$

  $$\begin{aligned} (r-{\tilde{r}}_{m})^3\ge (r-2r_{+})^3\ge (\frac{r_{0}}{8Q_{{\mathfrak {n}}}^{4}})^3, \end{aligned}$$

  and

  $$\begin{aligned} \begin{aligned}&\frac{180{\tilde{r}}_{m}Q_{{\mathfrak {n}}+1}\zeta }{(r-{\tilde{r}}_{m})^3} \le 180{\tilde{r}}_{m}Q_{{\mathfrak {n}}+1}\zeta (\frac{8Q_{{\mathfrak {n}}}^{4}}{r_0})^3 \le 180\frac{2r_0}{4Q_{{\mathfrak {n}}+1}^{4}}Q_{{\mathfrak {n}}+1}2\varepsilon _{0}^{\frac{1}{2}} \frac{512Q_{{\mathfrak {n}}}^{12}}{r_{0}^{3}}\\&\quad =C\frac{Q_{{\mathfrak {n}}}^{12}\varepsilon _{0}^{\frac{1}{2}}}{Q_{{\mathfrak {n}}+1}^{3}r_{0}^{2}} \le 1. \end{aligned} \end{aligned}$$

  (6.1)

  Thus, we have our conclusion $180{\tilde{r}}_{m}Q_{{\mathfrak {n}}+1}\zeta \le (r-{\tilde{r}}_{m})^3$.

• (2). From Lemma 2.1, we have $Q_{{\mathfrak {n}}+1}\ge Q_{{\mathfrak {n}}}^{{\mathcal {A}}}$ and $\ln Q_{{\mathfrak {n}}+1}\le Q^U_{\mathfrak {n}}.$ Using these and (5.20), we have

  $$\begin{aligned} \frac{256}{(r-{\tilde{r}}_m)^2}\textrm{e}^{-\frac{r-{\tilde{r}}_m}{2}Q_{{\mathfrak {n}}+1}}\zeta \le 256(\frac{8Q_{{\mathfrak {n}}}^{4}}{r_0})^{2} \textrm{e}^{-\frac{r-{\tilde{r}}_m}{2}Q_{{\mathfrak {n}}+1}}2\varepsilon _{0}^{\frac{1}{2}},\,\, \frac{256\cdot 64\cdot 2}{r_{0}^{2}}\varepsilon _{0}^{\frac{1}{2}}\le \frac{1}{2},\\ \begin{aligned} Q_{{\mathfrak {n}}}^{8}\textrm{e}^{-(r-{\tilde{r}}_m)Q_{{\mathfrak {n}}+1}}&\le \textrm{e}^{-\frac{r_0}{8Q_{{\mathfrak {n}}}^{4}}Q_{{\mathfrak {n}}+1}}Q_{{\mathfrak {n}}}^{8} \le \textrm{e}^{-\frac{Q_{{\mathfrak {n}}+1}}{Q_{{\mathfrak {n}}}^{5}}}Q_{{\mathfrak {n}}}^{8}\\&\quad \le \textrm{e}^{-\frac{Q_{{\mathfrak {n}}+1}^{\frac{1}{2}}\ln Q_{{\mathfrak {n}}+1}}{Q_{{\mathfrak {n}}}^{5}}}\textrm{e}^{Q_{{\mathfrak {n}}}\ln Q_{{\mathfrak {n}}+1}} \le \textrm{e}^{-\ln Q_{{\mathfrak {n}}+1}(Q_{{\mathfrak {n}}}^{\frac{{\mathcal {A}}}{2}-5}-Q_{{\mathfrak {n}}})}\\&\quad \le \textrm{e}^{-(\ln Q_{{\mathfrak {n}}+1})Q_{{\mathfrak {n}}}^{\frac{{\mathcal {A}}}{2}-5-1}} \le Q_{{\mathfrak {n}}+1}^{-({\mathfrak {n}}+2-n_{0})2^{{\mathfrak {n}}+2-n_{0}}c\tau U}\\&\quad = \left( \frac{\varepsilon }{\varepsilon _{-}}\right) ^{{\mathfrak {n}}+2-n_{0}} \le \varepsilon \le \tilde{\zeta }_m,\\ \end{aligned} \end{aligned}$$

  thus

  $$\begin{aligned} \frac{256}{(r-{\tilde{r}}_m)^2}e^{-\frac{r-{\tilde{r}}_m}{2}Q_{{\mathfrak {n}}+1}}\zeta \le \frac{1}{2}\tilde{\zeta }_m^{\frac{1}{2}}. \end{aligned}$$

• (3). We prove ${\tilde{K}}_m\le K^{(m)}$ and ${\tilde{K}}_m\le K.$ We first prove ${\tilde{K}}_m\le K^{(m)}.$ Owing to $\frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U}{60\tau }\ge 5$ and

  $$\begin{aligned} m\le L-1=1+\left\lfloor \frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U \ln Q_{{\mathfrak {n}}+1}}{60\tau \ln \frac{5}{2}}\right\rfloor , \end{aligned}$$

  then we have the inequalities

  $$\begin{aligned} \begin{aligned} {\tilde{K}}_m&\le \frac{2}{\tilde{\sigma }_m}\ln \frac{1}{\tilde{\varepsilon }_{m-1}} \le \frac{5\cdot 2^{m+2}\cdot 4Q_{{\mathfrak {n}}+1}^{4}}{r_0}\ln \frac{1}{\varepsilon ^{(\frac{5}{4})^{m-1}}}\\&\quad \le \frac{160(\frac{5}{2})^{m-1}Q_{{\mathfrak {n}}+1}^{4}}{r_0}\ln \frac{1}{\varepsilon } \le \frac{160(\frac{5}{2})^{\lfloor \frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U \ln Q_{{\mathfrak {n}}+1}}{60\tau \ln \frac{5}{2}}\rfloor }Q_{{\mathfrak {n}}+1}^{4}}{r_0}\ln \frac{1}{\varepsilon }\\&\quad \le \frac{160Q_{{\mathfrak {n}}+1} ^{\frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U}{60\tau }+4}}{r_0}\ln \frac{1}{\varepsilon } \le Q_{{\mathfrak {n}}+1}^{\frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U}{60\tau }}\ln \frac{1}{\varepsilon }\\&\quad \le \left( \frac{1}{\varepsilon }\right) ^{\frac{1}{60\tau }}\left( \frac{1}{\varepsilon }\right) ^{\frac{1}{60\tau }} = \left( \frac{1}{\varepsilon }\right) ^{\frac{1}{30\tau }}, \end{aligned} \end{aligned}$$

  (6.2)

  and

  $$\begin{aligned}{} & {} 2^{L+2}\le \left( \frac{5}{2}\right) ^{\lfloor \frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U \ln Q_{{\mathfrak {n}}+1}}{60\tau \ln \frac{5}{2}}\rfloor }\le Q_{{\mathfrak {n}}+1}^{\frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U}{60\tau }},\\{} & {} \frac{1}{2^{m+2}}\ge \frac{1}{2^{L+2}}\ge Q_{{\mathfrak {n}}+1}^{-\frac{2^{{\mathfrak {n}}+1-n_{0}}c\tau U}{60\tau }}\ge \varepsilon ^{\frac{1}{120\tau }}. \end{aligned}$$

  We conclude from all these inequalities that

  $$\begin{aligned} \begin{aligned}&\frac{K^{(m)}}{{\tilde{K}}_m} \ge \left( \frac{\gamma ^{2}\tilde{\sigma }_m^{2}}{2C_0\tilde{\zeta }_{m}^{\frac{1}{2}}}\right) ^{\frac{1}{3\tau }} \cdot \varepsilon ^{\frac{1}{30\tau }} \ge \left( \frac{\gamma _{0}^{2}r_{0}^{2}}{(20\cdot 2^{m+2}Q_{{\mathfrak {n}}+1}^{4})^{2}\cdot 2C_{0}\cdot 2\varepsilon ^{\frac{1}{2}}}\right) ^{\frac{1}{3\tau }}\cdot \varepsilon ^{\frac{1}{30\tau }}\\&\quad \ge \left( \frac{\gamma _{0}^{2}r_{0}^{2}}{C_{0}Q_{{\mathfrak {n}}+1}^{8}}\right) ^{\frac{1}{3\tau }} \cdot \left( \frac{1}{\varepsilon }\right) ^{\frac{1}{6\tau }-\frac{1}{30\tau }-\frac{1}{120\tau }} \ge \left( \frac{\gamma _{0}^{2}r_{0}^{2}}{C_{0}Q_{{\mathfrak {n}}+1}^{8}}\right) ^{\frac{1}{3\tau }}\cdot Q_{{\mathfrak {n}}+1}^{2^{{\mathfrak {n}}+2-n_{0}} c\tau U\left( \frac{1}{10\tau }-\frac{1}{30\tau }-\frac{1}{120\tau }\right) }\\&\quad \ge \left( \frac{\gamma _{0}^{2}r_{0}^{2}}{C_{0}Q_{{\mathfrak {n}}+1}^{8}}\right) ^{\frac{1}{3\tau }}\cdot Q_{{\mathfrak {n}}+1}^{2^{{\mathfrak {n}}+2-n_{0}}8} \ge 1, \end{aligned} \end{aligned}$$

  i.e., ${\tilde{K}}_m\le K^{(m)}.$ In (6.2), we have obtained ${\tilde{K}}_m\le (\frac{1}{\varepsilon })^{\frac{1}{30\tau }}$. This together with $K\ge \left( \frac{\gamma ^{2}}{2\varepsilon ^{\frac{1}{2}}}\right) ^{\frac{1}{10\tau }}$ implies that

  $$\begin{aligned} \begin{aligned} \frac{{\tilde{K}}_m}{K} \le \frac{\left( \frac{1}{\varepsilon }\right) ^{\frac{1}{30\tau }}}{\left( \frac{\gamma ^{2}}{2\varepsilon ^{\frac{1}{2}}}\right) ^{\frac{1}{10\tau }}} \le \left( \frac{1}{\varepsilon }\right) ^{\frac{1}{30\tau }} \left( \frac{2\varepsilon ^{\frac{1}{2}}}{\gamma ^{2}}\right) ^{\frac{1}{10\tau }} \le \left( \frac{2\varepsilon ^{\frac{1}{6}}}{\gamma ^{2}}\right) ^{\frac{1}{10\tau }} \le 1, \end{aligned} \end{aligned}$$

  (6.3)

  i.e., ${\tilde{K}}_m\le K.$ Thus, we complete the proof of ${\tilde{K}}_m\le \min \{K, K^{(m)}\}.$

$\square$

1.2 Some basic inequalities

Lemma 6.1

([4]) Let $g: {\mathcal {I}}\rightarrow {\mathbb {R}}$ be $b+3$ times differentiable, and assume that (1) $\forall \sigma \in {\mathcal {I}}$ there exists $s\le b+2$ such that $g^{(s)}(\sigma )>B$. (2) There exists A such that $|g^{(s)}(\sigma )|\le A$ for $\forall \sigma \in {\mathcal {I}}$ and $\forall s$ with $1\le s\le b+3$. Define ${\mathcal {I}}_h\equiv \{\sigma \in {\mathcal {I}}: |g(\sigma )|\le h\},$ then

$$\begin{aligned} \frac{\textrm{meas}({\mathcal {I}}_h)}{\textrm{meas}({\mathcal {I}})}\le \frac{A}{B}2\left( 2+3+\cdots +(b+3)+ 2B^{-1}\right) h^{\frac{1}{b+3}}. \end{aligned}$$