Global Attractor for Coupled Beam Equations with Nonhomogeneous Boundaries Conditions

Abstract

We study the longtime behavior for thermoelastic coupled beam equations with strong damping and past history thermal memory utt+uxxxx+uxxxxt+θxx-M∫0lux2dxuxx=hx, in 0,l×R+, and θt-ιθxx-1-ι∫0∞ksθxxt-sds-uxxt+pθ=qx, with 1/2≤ι<1, where we consider the nonlinear boundary conditions u0,t=ux0,t=uxxl,t=0, uxxxl,t+uxxxt(l,t)-M∫0lux2dxux(l,t)=ful,t+gutl,t,θ0,t=θxl,t=0 and certain initial conditions. To this problem little is known with nonlinear boundary conditions. This system arises from a model of the nonlinear thermoelastic coupled vibration beam with a nonlinear damping acting on x=l.