We study the long-time asymptotic behaviour of viscosity solutions $u(x,~t)$ of the Hamilton-Jacobi equation $u_t(x, t)+ H(x, u(x, t),$ $Du(x, t))= 0$ in $\mathbb{T}^n× {(-∞, ∞)}$, where $H= H(x, u, p)$ is convex and coercive in p and non-decreasing on u, and establish the uniform convergence of u to an an asymptotic solution u∞ as $t~\to \text{ }\infty $. Moreover, u∞ is a viscosity solution of Hamilton-Jacobi equation $H(x, u(x), Du(x))= 0$.
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