Variable cosmological term $\Lambda(t)$

We present the case of time-varying cosmological term $\Lambda(t)$. The main idea arises by proposing that as in the cosmological constant case, the scalar potential is identified as $V(\phi)=2\Lambda$, with $\Lambda$ a constant, this identification should be kept even when the cosmological term has a temporal dependence, i.e., $V(\phi(t))=2\Lambda(t)$. We Use the Lagrangian formalism for a scalar field $\phi$ with standard kinetic energy and arbitrary potential $V(\phi)$ and apply this model to the Friedmann-Robertson-Walker (FRW)cosmology. Exact solutions of the field equations are obtained by a special ansatz to solve the Einstein-Klein-Gordon equation and a particular potential for the scalar field and barotropic perfect fluid. We present the evolution on this cosmological term with different scenarios.