Asymptotic Behavior Of Solutions To The Kirchhoff Type Equations With Memory

LetΩbe a bounded open domain in R n with smooth boundary, H=L~2(Ω).We consider non-degenerate Kirchhoff type equations with memory term on the Hilbert space HWhere A is a linear operator in H , M and g are positive real functions. When the initial energy associated with the equations is non-negative and small, we prove the global existence and uniqueness of the weak solutions. We also study the asymptotic behaviour of the solutions. Under suitable assumptions on the damping term and memory term we prove the exponential decay of the solutions. When the Initial energy is negative and decay term is small enough, we prove that the solution blows up at some finite time.