General Decay Estimate for a Weakly Dissipative Viscoelastic Suspension Bridge

Abstract

In this article, we investigate the following weakly dissipative plate equation:

$$\displaystyle u_{tt}(x,y,t) + \Delta ^{2} u(x,y,t) + \int _0^t g(t-s) u_{xx}(x,y,s)ds-u_{xxt} = 0,\ \ \Omega \times (0,T). $$

Under some mild conditions on the relaxation function g, we show that the solution energy has general decay estimate. We also give some examples to illustrate our result. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s inequality and the generalized Young inequality are used to establish the stability results.