General decay of solutions for a viscoelastic suspension bridge with nonlinear damping and a source term

Abstract

In this paper, we study the nonlinear viscoelastic plate equation modeling the deformation of a suspension bridge, given by

$$\begin{aligned} u_{tt}(x,y,t)+\Delta ^2 u(x,y,t) -\displaystyle \int \limits _0^{t} g(t-\tau )\Delta ^2 u(\tau )\;\hbox {d}\tau +a(x, y) \vert u_t\vert ^m u_t+\vert u\vert ^{\rho }u=0, \end{aligned}$$

in the bounded domain $\Omega = (0, \pi )\times (-d, d)$, which is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. We prove, with general conditions on the relaxation function, that the decay rate of energy is similar to that of the relaxation function regardless of the presence or the absence of the frictional damping.