If $\Vert f(t,x)\Vert_{W^{1,\infty}((0,\infty)\times \mathbb{R})}\le C$, what conditions imply $f(t,x) \to g(x)$ uniformly as $t \to \infty$?

Question

Let $f:[0,\infty)\times \mathbb{R} \to \mathbb{R}$. Suppose that $$\Vert f(t,x)\Vert_{W^{1,\infty}((0,\infty)\times \mathbb{R})}\le C,$$ where $C$ is a constant. What conditions imply $$f(t,x) \to g(x) \text{ uniformly as } t \to \infty, $$ where $g:\mathbb{R} \to \mathbb{R}$ is independent of $t$?