Exponential decay of the solutions of the beams system
The paper proves the existence of the solution of the following abstract beams system: \[ u_{tt}+ A^2 u+ M(|A^{1/2}u|^2+|A^{1/2} v|^2) Au+ u_t= f,\tag{1} \] \[ v_{tt}+ A^2v+ M(|A^{1/2}u|^2+|A^{1/2} v|^2) Av+ v_t= g\tag{2} \] under the initial conditions \[ u(0)= u_0,\quad u_t(0)= u_1;\quad v(0)= v_0,\quad v_t(0)= v_1.\tag{3} \] Here, \(A\) denotes a positive selfadjoint operator of a Hilbert space with norm \(|\cdot|\), \(A^{1/2}\) is the square root of \(A\), \(M:[0,\infty)\to \mathbb{R}\) is a real function locally Lipschitz with certain properties, \(f\), \(g\) are given functions, \(u_0\), \(u_1\), \(v_0\), \(v_1\) are given constants. A weak formulation of the system (1), (2) with initial conditions (3) is presented and the pair of solutions \(u\), \(v\) is obtained as a limit of certain pairs of net functions \(u^\varepsilon\), \(v^\varepsilon\). The exponential decay of the energy \[ E(t)= \textstyle{{1\over 2}} \{|u_t|^2+|v_t|^2+|Au|^2+|Av|^2\} \] is proven.
scientific article; zbMATH DE number 1529450