Generalized comparison principle for contact Hamilton-Jacobi equations

In this paper, we discuss all the possible pairs $(u,c)\in C(M,\mathbb{R})\times \mathbb{R}$ solving (in the sense of viscosity) the contact Hamilton-Jacobi equation $$H(x,d_{x}u,u)=c,x\in M$$ of which $M$ is a closed manifold and the continuous Hamiltonian $H:(x,p,u)\in T^{\ast }M\times \mathbb{R}\to \mathbb{R}$ is convex, coercive in $p$ but merely non-decreasing in $u$. Firstly, we propose a comparison principle for solutions by using the dynamical information of Mather measures. We then describe the structure of $\mathfrak{C}$ containing all the $c\in \mathbb{R}$ makes previous equation solvable. We also propose examples to verify the optimality of our approach.