Timeline for Equivalence of statements about level sets: $u|_{S \times [\tau, \infty)}$ depends only on $t$ $\iff$ $u(t,x) = \mu^{\tau}(t,u(\tau,x))$

8 events

when toggle format	what		by	license	comment
Apr 12, 2021 at 12:46	answer	added	mlk		timeline score: 1
Apr 12, 2021 at 12:39	comment	added	Riku		@mlk Thank you. Yes, there was a typo in the first line too: it was $u(\color{red}{\tau}, \cdot)$. Could you expand on your comment with more detail in an answer?
Apr 12, 2021 at 12:36	history	edited	Riku	CC BY-SA 4.0	added 3 characters in body
Apr 12, 2021 at 11:23	comment	added	mlk		I assume that in 1. you also mean level sets of $u(\tau,.)$, (otherwise the "depends only on $t$" is a bit circular). With that correction, the statement looks true on the level of sets (no need for any smoothness), as 1. implies that $u(t,x)$ is independent of $x$ on any given level set which is enough to define $\mu^\tau$ and prove 2. and conversely on any level set $S$, the function $\mu^\tau$ only depends on $t$ which implies 1.
Apr 12, 2021 at 10:13	comment	added	Riku		@mlk There was a typo. The line is $u(t,x) = \mu^{\tau}(t,u(\color{red}{\tau},x))$
Apr 12, 2021 at 10:13	history	edited	Riku	CC BY-SA 4.0	added 3 characters in body
Apr 12, 2021 at 6:39	comment	added	mlk		Is there some derivative missing in condition 2? As it is stated now, I could just write $\mu^\tau(t,y):=y$ to make it trivially true, even if condition 1 is not.
Apr 11, 2021 at 23:19	history	asked	Riku	CC BY-SA 4.0