2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225304In this work we consider the Navier-Stokes problem modified by the absorption term $|\textbf{u}|^{σ-2}\textbf{u}$, where $σ>1$, 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 $1<σ<2$, exponentially decay in time if $σ=2$ and decay with a power-time rate if $σ>2$. % We prove also that for a general non-zero body forces, the weak solutions exponentially decay in time for any $σ>1$. 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 $1<σ<2$.22 pagesAnalysis of PDEsMathematical Physics35Q30, 76D03, 35B40text