Pazoto, A. F.; Perla Menzala, G. Uniform stabilization of a nonlinear beam model with thermal effects and nonlinear boundary dissipation. (English) Zbl 1142.35615 Funkc. Ekvacioj, Ser. Int. 43, No. 2, 339-360 (2000). From the introduction: We consider the solution-pair \({u(x,t),\theta(x,t)}\) of the following initial-boundary value problem \[ u_{tt}+u_{xxxx}-u_{xxtt}-M(\int_0^Lu_x^2dx)u_{xx}+\alpha \theta_{xx}=0, \]\[ \theta_t-\theta_{xx}-\alpha u_{xxt}=0, \] in \(0<x<L\), \(t>0\), with the boundary conditions \[ u(0,t)=u_x(0,t)=\theta(0,t)=0, \]\[ \theta_x(L,t)+\lambda\theta(L,t)=0, \]\[ (u_{xx}+u_{xt}+\alpha\theta)(L,t)=0, \]\[ (u_{xxx}-u_{xxt}-M(\int_0^Lu_x^2 dx)u_x +\alpha\theta_x)(L,t)=f(u_t(L,t)), \] for \(t>0\) and initial conditions \[ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad\theta(x,0)=\theta_0(x), \] in \(0<x<L\).Our main result in this paper says that \(E(t)\) decays to zero at a uniform rate as \(t\to\infty\) on bounded sets of initial data. Cited in 11 Documents MSC: 35Q72 Other PDE from mechanics (MSC2000) 74H20 Existence of solutions of dynamical problems in solid mechanics 74H25 Uniqueness of solutions of dynamical problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 93C20 Control/observation systems governed by partial differential equations 93D15 Stabilization of systems by feedback PDFBibTeX XMLCite \textit{A. F. Pazoto} and \textit{G. Perla Menzala}, Funkc. Ekvacioj, Ser. Int. 43, No. 2, 339--360 (2000; Zbl 1142.35615)