Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?

1. For every $\tau >0$ and level surface $S$ of $u(t,\cdot)$, it holds that $u|_{S \times [\tau, \infty)}$ depends only on $t$
2. For every $\tau >0$, there exists $\mu^\tau: [\tau, \infty) \times \mathbb R \to \mathbb R$ such that $u(t,x) = \mu^{\tau}(t,u(\tau,x))$

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?

1. For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$, it holds that $u|_{S \times [\tau, \infty)}$ depends only on $t$
2. For every $\tau >0$, there exists $\mu^\tau: [\tau, \infty) \times \mathbb R \to \mathbb R$ such that $u(t,x) = \mu^{\tau}(t,u(\tau,x))$

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?

1. For every $\tau >0$ and level surface $S$ of $u(t,\cdot)$, it holds that $u|_{S \times [\tau, \infty)}$ depends only on $t$
2. For every $\tau >0$, there exists $\mu^\tau: [\tau, \infty) \times \mathbb R \to \mathbb R$ such that $u(t,x) = \mu^{\tau}(t,u(t,x))$

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?

1. For every $\tau >0$ and level surface $S$ of $u(t,\cdot)$, it holds that $u|_{S \times [\tau, \infty)}$ depends only on $t$
2. For every $\tau >0$, there exists $\mu^\tau: [\tau, \infty) \times \mathbb R \to \mathbb R$ such that $u(t,x) = \mu^{\tau}(t,u(\tau,x))$