Stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation
The authors consider the viscoelastic Euler-Bernoulli type initial-boundary value problem with a localized damping term: \[ u_{tt}+\Delta^2u-M(\| u\|^2)\Delta u- \int_0^tg(t-\tau)\Delta^2u(\tau)\,d\tau+\rho(x,u_t)=0\quad \text{in }\Omega\times \mathbb{R}_+, \] \[ u=\frac {\partial u}{\partial \nu}=0\quad \text{on }\Gamma\times \mathbb{R}_+; \qquad u(x,0)=u_0(x), \quad u'(x,0)=u_1(x),\qquad x\in \Omega, \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(n\geq 1\), with a boundary \(\Gamma=\Gamma_0\cup \Gamma_1\) of the \(C^2\) class, where \(\Gamma_0\) and \(\Gamma_1\) are closed and disjoint, \(\Gamma_0\neq \emptyset\) and satisfying the conditions \( m\cdot \nu\geq 0>0\) on \(\Gamma_1\), \(m\cdot \nu\leq 0\) on \(\Gamma_0\), \(m(x)=x-x_0\) \((x_0\in \mathbb{R}^n)\). Further \(M\in C^1(\mathbb{R}_+)\), \(g:\mathbb{R}_+\to \mathbb{R}_+\) and \(\rho(x,s)\) is almost everywhere differentiable in \(x\) and nondecreasing in \(s\). The work is devoted to prove the existence of global solutions and decay for the energy of solutions. Due to my opinion the assumptions for the boundary cannot be fulfilled for the \(C^2\) boundary. The strong convergence (3.22) is wrong, in order to pass to the limit in the equation it suffices the strong convergence in \(C([0,T];H^1_0(\Omega))\) instead of \(C([0,T];V)\), \(V=H^2_0(\Omega)\).
scientific article; zbMATH DE number 5667598