  We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups,
with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From
this study we derive the asymptotic behaviour of a large class of linear
Fokker-Plank type equations under simple conditions, widely extending previous
results. Nonlinear diffusion equations are also studied by means of these
inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery appears as a
main tool in the analysis, in local or integral forms.

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