General decay of solutions for a von Karman plate system with general type of relaxation functions on the boundary

Abstract

In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value $ w_0 \equiv 0 $ on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels $ k_i(i = 1, 2) $, namely $ k_i''(t) \geq - \xi_i(t) G_i(-k_i'(t)) $, where $ G_i $ are convex and increasing functions near the origin and $ \xi_i $ are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions $ \xi_i $ and $ G_i. $ These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature.

Keywords

• von karman plate
• general decay
• memory term
• relaxation function
• convexity
• boundary condition