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A Nekhoroshev type theorem for the nonlinear wave equation. (English) Zbl 1444.37058

The authors consider the nonlinear wave equation \[ \displaystyle{u_{tt} = u_{xx}-mu - f(u),}\tag{1} \] with Dirichlet periodic boundary conditions \[ u(t,0)=u(t,\pi)=0,\tag{2} \] with \(f\) being analytical, \(0<m_0\leq m\leq \Delta\); \(m_0\) and \(\Delta\) are given. The main problem here is the study of the long-time stability, i.e., for \(|t|<c\epsilon^{-r+1/2}\) with \(\epsilon>0\) small enough. Since the topology of the underlying system is important for such property, the authors consider the Gevrey space defined by \[ \displaystyle{G_\sigma = \left\{z = \{z_j\}_{j\in \bar{Z}} \, :\, \sum_{j\in\bar{Z}}|z_j|e^{\sigma\sqrt{|j|}}<\infty \right\}} \] with \(\sigma>0\), \(\bar{Z}=Z\setminus\{0\}\). The solution of the problem is expanded as \[ \displaystyle{u(t,x)=\sum_{j\geq 1}\frac{u_j(t)}{\sqrt{\omega_j}}\phi_j\;,\;v(t,x)=u_t(t,x) = \sum_{j\geq 1}\sqrt{\omega_j}v_j(t)\phi_j,} \] where \[ \displaystyle{\phi_j=\sqrt{2/\pi}\sin\;jx\;,\;\omega_j^2=j^2+m\;;\;z_j(t)=\frac{u_j(t)}{\sqrt{\omega_j}}\;,\;z_{-j}(t)=\sqrt{\omega_j}v_j(t).} \]
The main result of the paper is as follows.
Theorem. Given any \(0<m_0<\Delta\), \(0<\beta<1/6\), \(\sigma>0\) and \(\rho>0\) there exist constants \(c>0\), \(\epsilon_0>0\) such that if \[ \displaystyle{\sum_{j\in\bar{Z}}|z_j(0)|e^{\sigma\sqrt{|j|}+\rho|j|}=\epsilon<\epsilon_0} \] the solution of (1)-(2) with initial data \[ \displaystyle{u(0,x)=\sum_{j\geq 1}z_j(0)\phi_j\ ,\ v(0,x)=\sum_{j\geq 1}z_{-j}(0)\phi_j} \] satisfies the following inequality \[ \displaystyle{\sum_{j\in\bar{Z}}|z_j(t)|e^{\sigma\sqrt{|j|}+\rho|j|}\leq c\epsilon\;,\;|t|\leq e^{-\zeta|\ln\;\epsilon|^\beta},} \] with \(\zeta=\min\{1/4\;,\;(2-\sqrt{2})\sigma\}\).

MSC:

37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35B35 Stability in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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References:

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