EXISTENCE, UNIQUENESS AND EXPONENTIAL DECAY OF SOLUTIONS TO KIRCHHOFF EQUATIONIN R-n

Abstract

We discuss the global well-posedness and uniform exponential stability for the Kirchhoff equation in R-n u(tt) - M(integral(Rn) vertical bar Delta u vertical bar(2)dx)Delta u + lambda(ut) = 0 in R-n x (0, infinity). The global solvability is proved when the initial data are taken small enough and the exponential decay of the energy is obtained in the strong topology H-2(R-n) x H-1(R-n), which is a different feature of the present article when compared with the prior literature. We also dedicate a section to discuss a model with the frictional damping term lambda(ut), is replaced by a viscoelastic damping term integral(t)(0) g(t - s)Delta u(s)ds..