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

The objective of this paper is to present some results about viscosity subsolutions of the contact Hamiltonian-Jacobi equations on connected, closed manifold $M$ $$H(x,{\mathrm{\partial }}_{x}u,u)=0,x\in M\mathrm{.}$$ Based on implicit variational principles introduced in [12,14], we focus on the monotonicity of the solution semigroups on viscosity subsolutions and the positive invariance of the epigraph for viscosity subsolutions. Besides, we show a similar consequence for strict viscosity subsolutions on $M$.