Viscosity solutions of contact Hamilton-Jacobi equations without monotonicity assumptions.

This paper deals with: (1) the generalized additive eigenvalue problem \[ H(x,u(x),Du(x))=c, \quad x\in M, \] where the unknown is a pair $(c,u)$ of a constant $c \in \mathbb{R}$ and a function $u$ on $M$ for which $u$ is a backward weak KAM solution (or equivalently, viscosity solution) of the above equation; (2) the long-time behavior of viscosity solutions of the Cauchy problem for \[ w_{t}(x,t)+H(x, w(x,t), Dw(x,t))=c,\quad (x,t)\in M\times(0,+\infty). \] We assume $H=H(x,u,p)$ satisfies Tonelli conditions in the argument $p\in T^*_xM$ and the uniform Lipschitz condition in the argument $u\in\mathbb{R}$. First, we provide three necessary and sufficient conditions for the existence of backward weak KAM solutions of the stationary equation for a given $c\in \mathbb{R}$. Second, we analyse the structure of the set $\mathfrak{C}$ of all real numbers $c$'s for which the stationary equation admits backward weak KAM solutions. Third, the long-time behavior of viscosity solutions of the Cauchy problem for the evolutionary equation is studied. Implicit Lax-Oleinik semigroups $\{T^\pm_t\}_{t\geqslant 0}$ play an essential role in this paper. The most important novelty in this work is to reveal several new aspects of existence and long-time behavior of viscosity solutions of the above equations without monotonicity assumptions $\frac{\partial H}{\partial u}\geqslant 0$ (or, $\frac{\partial H}{\partial u}\leqslant 0$): the propeties of the operator $T^+_t\circ T^-_t$ is used to provide a necessary and sufficient condition for the existence of viscosity solutions of the stationary equation; three constants are introduced to analyse the structure of $\mathfrak{C}$; the information of $w$ in finite time partly determines the long-time behavior of $w$.