Suppose $u: \mathbb{R}^m \times [0,\infty) \to \mathbb{R}$ and $u_0,g: \mathbb{R}^m \to \mathbb{R}$ where $m$ is 2 or 3. Further, suppose that $u_0$ and $g$ are both smooth. Let's say $u(x,t)$ is a solution (maybe in the sense of viscosity solution) to the initial value problem:

\begin{align} u_t = g\|Du\|, \;\; & (x,t) \in \mathbb{R}^m \times (0, \infty) \\ u(x,t) = u_0(x), \;\; & (x,t) \in \mathbb{R}^m \times \{0\} \end{align}

Intuitively, because $u$ is non-decreasing in $t$ where $g > 0$ and non-increasing where $g < 0$, it seems that we should have $$ \text{sign}(u) \to \text{sign}(g), \;\; t \to \infty \tag{1} $$ so long as (where defined) $\|Du\| \nrightarrow 0$, or, if the gradient norm does go to zero, at least not too quickly.

Is there literature on whether the this conjecture is true under these conditions, or perhaps either more strict or more general ones?

**Edit**:

To be a little more specific:

We should exclude in the possibility $u_0(x) = \text{constant}$, since clearly $u(x,t)$ won't move if this is the case. I'm interested in a set of restrictions on $u_0$ and $g$ (besides $u_0=g$) such that $(1)$ is guaranteed.