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Our aim in this article is to study a nonlinear viscoelastic Kirchhoffequation with strong damping, Balakrishnan-Taylor damping, nonlinear source anddynamical boundary condition. Firstly, we prove the local existence ofsolutions by using the Faedo-Galerkin approximation method combined with acontraction mapping theorem. We then prove that if the initial data enter intothe stable set, the solution globally exists, and if the initial data enterinto the unstable set, the solution blows up in a finite time. Moreover, weobtain a general decay result of the energy, from which the usual exponentialand polynomial decay rates are only special cases.

Existence, general decay and blow-up of solutions for a viscoelastic  Kirchhoff equation with Balakrishnan-Taylor damping and dynamic boundary  conditions