In the context of partial differential equations, in the book I'm reading, to formulate uniqueness of the solution of the one dimensional heat equation the author makes the following definition:
We say that $u(x,t)$ belongs to $\mathcal{S}(\mathbb{R})$ uniformly in $t$ if for any $T > 0$ we have
$$\sup_{\ \ \ x\in \mathbb{R} \\ 0<t<T} |x|^k \left|\dfrac{\partial^l}{\partial x^l}u(x,t)\right|<\infty, \quad \text{for each $k,l\geq 0$}$$
Now all this says is that fixed $k,l\geq 0$ integers, there is $A_{k,l}\in \mathbb{R}$ such that
$$|x|^k \left|\dfrac{\partial^l}{\partial x^l}u(x,t)\right|\leq A_{k,l}$$
for all $x\in \mathbb{R}$ and $t\in (0,T)$ for every $T>0$.
Now I wanted some intuition, I wanted to really understand what this means.
So what is the intuition about this definition? How to understand what this actually means? Why this is important?
I imagine the point is to formulate the idea that for each $t$ fixed, the function $u(x,t)$ is an element of the Schwartz space. In other words, for each $t$ fixed we have $|x|^k |\partial_l u(x,t)|$ bounded. But this bound could depend on $t$. So it seems the definition is meant to formalize the idea that the bound doesn't depend on $t$.
What i don't understand is why do we require that for each $T > 0$ we have that $\sup$ property with $t\in (0,T)$. Why not already consider $t\in [0,\infty]$, or even $t\in \mathbb{R}$?
Imagine that the question were initially raised in the context of a normed vector space of functions, $(X, \Vert\cdot\Vert)$. In this case we might be interested not only in the qualitative inclusion $u(\cdot,t) \in X$ but also in the quantitative information of how big $u(\cdot,t)$ is when measured in $X$, i.e. $\Vert u(\cdot,t)\Vert$. There are all sorts of ways we could measure this due to the fact that $t \mapsto \Vert u(\cdot,t)\Vert$ is a function of one variable. For example, we could consider $$ \left(\int_0^T \Vert u(\cdot,t)\Vert^p dt\right)^{1/p} \text{ for } 1 \le p < \infty $$ or we could also consider $$ \sup_{0\le t \le T} \Vert u(\cdot,t) \Vert. $$ The latter is usually thought of as providing uniform control, for obvious reasons. Indeed, if we know that $$ \sup_{0\le t \le T} \Vert u(\cdot,t) \Vert = M < \infty $$ then we know that the trajectory $t \mapsto u(\cdot,t) \in X$ is confined to the closed ball $B_M$ for all $0\le t \le T$. We don't get this information from the integral terms above.
Now recall that the Schwartz class does not come equipped with a norm, so none of the above is directly applicable. However, it does come equipped with a family of seminorms, namely $$ [u]_{k,l} = \sup_{x} \vert x \vert^k \vert D^\ell u(x) \vert. $$ The upshot is that we don't have a single quantity to use to measure the "size" of a function in the Schwartz class, but we do have countably many, and this is almost as good. The definition you state is mimicking the uniform control above by applying the same idea to each of the seminorms in the family, namely requiring $$ \sup_{0\le t \le T} [u(\cdot,t)]_{k,l} = M_{k,l}. $$ This, of course, has the same interpretation as above: if we imagine $[\cdot]_{k,l}$ as generating a ball then $u(\cdot,t)$ is confined to this ball.
I must admit I don't have a great answer for your second question. If it were up to me I would not use the expression "belongs uniformly" for what is written in your post. I would define belongs uniformly to mean something like $$ \sup_{0\le t <\infty} [u(\cdot,t)]_{k,l} = M_{k,l} \text{ for each }k,l, $$ and rename the given definition as "belongs locally uniformly." Here the operative word is "locally," which is often employed to mean that something holds when restricted to compact sets. For instance, we say a function $f: \mathbb{R} \to \mathbb{R}$ is locally integrable, or $f \in L^1_{loc}(\mathbb{R})$ if the restriction of $f$ to each compact set $K \subset \mathbb{R}$ is integrable. This seems to be what's going on in the definition from your post: the uniform inclusion is required to hold on each compact set $[0,T]$, but the constants $M_{k,l}$ depend on $T$ and can blow up as $T \to \infty$. As with any definition, what really matters is how it's used, so I would guess that this local uniform control is good enough for whatever the author wants to use it for. For example, it will be useful if you have to integrate $u$ in time over compact intervals.