The generalized three dimensional Navier–Stokes equations with damping are considered. Firstly, existence and uniqueness of strong solutions in the periodic domain \({\mathbb {T}}^{3}\) are proved for \(\frac{1}{2}<\alpha <1,~~ \beta +1\ge \frac{6\alpha }{2\alpha -1}\in (6,+\infty )\) . Then, in the whole space \(R^3,\) if the critical situation \(\beta +1= \frac{6\alpha }{2\alpha -1}\) and if \(u_{0}\in H^{1}(R^{3}) \bigcap {\dot{H}}^{-s}(R^{3})\) with \(s\in [0,1/2]\) , the decay rate of solution has been established. We give proofs of these two results, based on energy estimates and a series of interpolation inequalities, the key of this paper is to give an explanation for that on the premise of increasing damping term, the well-posedness and decay can still preserve at low dissipation \(\alpha <1,\) and the relationship between dissipation and damping is given.
