The Navier-Stokes problem modified by an absorption term

In this work we consider the Navier-Stokes problem modified by the absorption term $|\textbf{u}|^{\sigma-2}\textbf{u}$, where $\sigma>1$, which is introduced in the momentum equation. % For this new problem, we prove the existence of weak solutions for any dimension $N\geq 2$ and its uniqueness for N=2. % Then we prove that, for zero body forces, the weak solutions extinct in a finite time if $1<\sigma<2$, exponentially decay in time if $\sigma=2$ and decay with a power-time rate if $\sigma>2$. % We prove also that for a general non-zero body forces, the weak solutions exponentially decay in time for any $\sigma>1$. In the special case of a suitable forces field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided $1<\sigma<2$.