On global small solutions of nonlinear Timoshenko's type equations (Q1293158)

The nonlinear Timoshenko's type equation \[ u_{tt}+ a(\|D^{\beta_1}u\|^2_{L^2},\dots,\|D^{\beta_N}u\|^2_{L^2}) \Delta^2u= m(\|D^{\beta_1}u\|^2_{L^2},\dots,\|D^{\beta_N}u \|^2_{L_2})\Delta u \] is appearing in various models studying the nonlinear vibrations of beams and plates. The considered paper is devoted to the global existence of its small, regular solutions in \(\mathbb{R}_t\times \mathbb{R}^n_x\), when the functions \(a,m\in C^1([-\delta, \delta]^N)\) satisfy a strict hyperbolicity condition \(a(s),m(s)\geq \eta>0\) for all \(s\in [-\delta, \delta]^N\). The multi-indces \(\beta_v= (\beta_{v,t}, \beta_{v,x})\in \mathbb{N}\times \mathbb{N}^n\) are such that \(|\beta_{v,t}|\leq 1\), \(|\beta_v|\geq 1\) for \(1\leq v\leq N\). For \(n\geq 2\) it is proved that the Cauchy problem for the above equation with the conditions \(u(0,x)= \varepsilon u_0(x)\), \(u_t(0, x)=\varepsilon u_1(x)\), \(u_0(x),u_1(x)\in C^\infty_0(\mathbb{R}^n_x)\), has a unique classical solution \(u\in C^2(\mathbb{R}_t; H^\infty(\mathbb{R}^n_x))\) provided the parameter \(\varepsilon\) is small enough. For \(n=1\) there is a remark that the above statement could be proved only in the case \(m(s_1,\dots, s_N)= \mu a(s_1,\dots, s_N)\) for some \(\mu>0\).