Existence and sharp decay rate estimates for a von Karman system with long memory

A nonlinear model described by von Karman equations with long memory is considered. Hadamard wellposedness of weak solutions, regularity of solutions and intrinsic decay rate estimates for the energy are established by assuming that the memory kernel gg satisfies the inequality introduced in Alabau-Boussouira and Cannarsa (2009): g′≤−H(g)g′≤−H(g), where H(s)H(s) is a given continuous, positive, increasing, and convex function such that H(0)=0H(0)=0. The decay rates obtained are optimal in the sense that they reconstruct decay rates assumed on relaxation kernel. The novelty of the paper is at the level of both: the results obtained and the methodology applied.