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Uniform stabilization of a nonlinear beam model with thermal effects and nonlinear boundary dissipation. (English) Zbl 1142.35615

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.

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
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