An interacting dark energy model with nonminimal derivative coupling

We study cosmological dynamics of an extended gravitational theory that gravity is coupled non-minimally with derivatives of a dark energy component and there is also a phenomenological interaction between the dark energy and dark matter. Depending on the direction of energy flow between the dark sectors, the phenomenological interaction gets two different signs. We show that this feature affects the existence of attractor solution, the rate of growth of perturbations and stability of the solutions. By considering an exponential potential as a self-interaction potential of the scalar field, we obtain accelerated scaling solutions that are attractors and have the potential to alleviate the coincidence problem. While in the absence of the nonminimal derivative coupling there is no attractor solution for phantom field when energy transfers from dark matter to dark energy, we show an attractor solution exists if one considers an explicit nonminimal derivative coupling for phantom field in this case of energy transfer. We treat the cosmological perturbations in this setup with details to show that with phenomenological interaction, perturbations can grow faster than the minimal case.