Almost Global Existence for d-dimensional Beam Equation with Derivative Nonlinear Perturbation

Abstract

This paper is devoted to the proof of almost global existence results for the d-dimensional beam equation with derivative nonlinear perturbation by using Birkhoff normal form technique and the so-called tame property in a Gevrey space.

Appendix

1.1 Technical Lemma

Lemma 7.1

For any $s>\frac{3}{2}$, $\sigma >\rho \ge 0$ and any ${\varvec{j}},{\varvec{l}}\in {\mathbb {Z}}^{d}$, we assume that

$$\begin{aligned} \psi ({\varvec{j}},s,\sigma )\le \psi ({\varvec{j}}-{\varvec{l}},s,\sigma )\cdot \psi ({\varvec{l}},s,\rho )\qquad \text{ when }\ \left| {\varvec{j}}-{\varvec{l}}\right| _2\ge \left| {\varvec{l}}\right| _2 \end{aligned}$$

(57)

and

$$\begin{aligned} \psi ({\varvec{j}},s,\sigma )\le \psi ({\varvec{j}}-{\varvec{l}},s,\rho )\cdot \psi ({\varvec{l}},s,\sigma )\qquad \text{ when }\ \left| {\varvec{j}}-{\varvec{l}}\right| _2< \left| {\varvec{l}}\right| _2, \end{aligned}$$

(58)

with $\psi ({\varvec{j}},s,\sigma )=|{\varvec{j}}|_2^{s}e^{\sigma \sqrt{\left| {\varvec{j}}\right| _2}}$. Then there exists $\sigma '$ satisfying $\sigma>\sigma '>\rho$ such that

$$\begin{aligned} \left\| q*q'\right\| _{s,\sigma }\le C(s, \sigma ',\rho )\left( \left\| q\right\| _{s,\sigma }\left\| q'\right\| _{s,\sigma '}+\left\| q\right\| _{s,\sigma '} \left\| q'\right\| _{s,\sigma }\right) , \end{aligned}$$

(59)

where $q*q'$ is the convolution defined by

$$\begin{aligned} (q*q')_{{\varvec{j}}}=\sum _{{\varvec{l}}\in {\mathbb {Z}}^d}q_{{\varvec{j}}-{\varvec{l}}}q'_{{\varvec{l}}} \end{aligned}$$

(60)

and $C(s,\sigma ',\rho )>0$ is a constant depending on $s, \sigma '$ and $\rho$ only.

Proof

In view of (57), (58) and (60), one has

$$\begin{aligned} \left\| q*q'\right\| _{s,\sigma }^{2}\le & {} \sum _{{\varvec{j}}\in {\mathbb {Z}}^{d}}\left| \sum _{{\varvec{l}}\in {\mathbb {Z}}^{d}}q_{{\varvec{j}}-{\varvec{l}}}\cdot q'_{{\varvec{l}}}\right| ^{2}{\psi ({\varvec{j}}-{\varvec{l}},s, \sigma )^2\cdot \psi ({\varvec{l}},s,\rho )^2} \\{} & {} +\sum _{{\varvec{j}}\in {\mathbb {Z}}^{d}}\left| \sum _{{\varvec{l}}\in {\mathbb {Z}}^{d}}q_{{\varvec{j}}-{\varvec{l}}}\cdot q'_{{\varvec{l}}}\right| ^{2}{\psi ({\varvec{j}}-{\varvec{l}},s, \rho )^2\cdot \psi ({\varvec{l}},s,\sigma )^2}\\\le & {} C(s,\sigma ',\rho )\left( \left\| q\right\| _{s,\sigma }\left\| q'\right\| _{s,\sigma '}+\left\| q\right\| _{s,\sigma '} \left\| q'\right\| _{s,\sigma }\right) , \end{aligned}$$

where the last inequality uses Young inequality

$$\begin{aligned} \left\| a*b\right\| _{\ell ^2}\le \left\| a\right\| _{\ell ^{2}}\left\| b\right\| _{\ell ^{1}},\quad a\in \ell ^{2}, b\in \ell ^{1}, \end{aligned}$$

and Cauchy inequality. $\square$

Remark 7.2

In view of Lemma 7.1 and Lemma 2.2 in [11], one gets

$$\begin{aligned} \rho = (\sqrt{2}-1)\sigma . \end{aligned}$$

(61)

Lemma 7.3

Given two positive integers m, n with $m\ge n+3$, let $\ell _i\in \{3,4,\dots ,m-n\}$ for $1\le i\le n+1$. Then one has

$$\begin{aligned} \max _{\ell _1+\cdots +\ell _{n+1}=m+2n}\left( \ell _1^6+\cdots +\ell _{n+1}^6\right) =3^{6}n+(m-n)^6. \end{aligned}$$

(62)

Proof

Firstly, assume $b\ge a\ge 4$, then it is easy to see that

$$\begin{aligned} a^{6}+b^6\le (a-1)^{6}+(b+1)^{6}\le \cdots \le 3^6+\left( b+a-3\right) ^6. \end{aligned}$$

(63)

Using (63) again and again, we have

$$\begin{aligned} \ell _1^6+\cdots +\ell _{n+1}^6&\le 3^6+\ell _{2}^6+\cdots +\left( \ell _{n+1}+(\ell _1-3)\right) ^6\\&\le 3^6\cdot 2+\ell _{3}^6+\cdots +(\ell _{n+1}+(\ell _1-3)+(\ell _2-3))^6\\&\quad \cdots \\&\le 3^6n+(m-n)^6, \end{aligned}$$

where the last inequality uses $\ell _1+\cdots +\ell _{n+1}=m+2n$. $\square$

Lemma 7.4

Let $u^{(1)}, \ldots , u^{(K)}$ be K independent vectors with $|u^{(i)}|\le 1$. Let $w\in {\mathbb {R}}^{K}$ be an arbitrary vector, then there exists $i\in \{1,\ldots ,K\},$ such that

$$\begin{aligned} |u^{(i)}\cdot w|\ge \frac{|w|_{1}\det (u^{(i)})}{K^{3/2}}, \end{aligned}$$

where $\det (u^{(i)})$ is the determinant of the matrix formed by the components of the vectors $u^{(i)}$.

Proof

The proof can be found in Lemma 5.2 of [3]. $\square$

Lemma 7.5

Suppose that g(m) is r times differentiable on an interval $J\subset {\mathbb {R}}$. Let $J_{\varrho }:=\{m\in J: |g(m)|<\varrho \},\;\varrho >0.$ If $|g^{(r)}(m)|\ge \delta _*>0$ on J, then

$$\begin{aligned} \text{ Meas }\, J_{\varrho }\le M\varrho ^{1/r}, \end{aligned}$$

where $M:=2(2+3+\cdots +r+\delta _*^{-1}).$

Proof

The proof can be found in Lemma 5.4 of [3]. $\square$