First, we provide a necessary and sufficient condition of the existence of viscosity solutions of the nonlinear first order PDE$F(x,u,Du)=0,x\in M,$ under which we prove the compactness of the set of all viscosity solutions. Here, $F(x,u,p)$ satisfies Tonelli conditions with respect to the argument p and $-\lambda \le \frac{\partial F}{\partial u}<0$ for some $\lambda >0$, and M is a compact manifold without boundary. Second, we study the long time behavior of viscosity solutions of the Cauchy problem for$w_t+F(x,w,w_x)=0,(x,t)\in M\times (0,+\infty ),$ from the weak KAM point of view. The dynamical methods developed in [13], [14], [15] play an essential role in this paper.

中文翻译：

---

依赖未知函数的哈密顿-雅各比方程的弱KAM解

首先，我们提供了非线性一阶PDE粘度解存在性的充要条件$F（X，ü，dü）=0，X\in \mathrm{中号}，$在此之下，我们证明了所有粘度溶液组的紧致性。这里，$F（X，ü，p）$关于参数p和t满足Tonelli条件$-\lambda \le \frac{\partial F}{\partial ü}<0$ 对于一些 $\lambda >0$，M是一个无边界的紧流形。其次，我们研究了柯西问题的粘性溶液的长时间行为$w_{Ť}+F（X，w，w_X）=0，（X，Ť）\in \mathrm{中号}\times （0，+\infty ），$从弱KAM的角度来看。在[13]，[14]，[15]中开发的动力学方法在本文中起着至关重要的作用。