Initial Boundary Value Problem for a Fractional Viscoelastic Equation of the Kirchhoff Type

In this paper, we study the initial boundary value problem for a fractional viscoelastic equation of the Kirchhoff type. In suitable functional spaces, we define a potential well. In the framework of the potential well theory, we obtain the global existence of solutions by using the Galerkin approximations. Moreover, we derive the asymptotic behavior of solutions by means of the perturbed energy method. Our main results provide sufficient conditions for the qualitative properties of solutions in time.