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The objective of this paper is to present some results about viscositysubsolutions of the contact Hamiltonian-Jacobi equations on connected, closedmanifold $M$ $$ H(x,\partial_x u,u)= 0, \quad x\in M. $$ Based on implicitvariational principles introduced in [12,14], we focus on the monotonicity ofthe solution semigroups on viscosity subsolutions and the positive invarianceof the epigraph for viscosity subsolutions. Besides, we show a similarconsequence for strict viscosity subsolutions on $M$.

Viscosity subsoltions of Hamilton-Jacobi equations and Invariant sets of contact Hamilton systems