Let $f(t,x)$ be any given function (solution of some partial differential equation) for any $(t,x)\in[0,T]\times [0,1].$
I want to know what is the difference between the spaces $C([0,T],L^p[0,1])$ and $L_{\infty}([0,T],L^p[0,1])$ containing the function $f(t,x).$ And what is the relation between these two spaces?
More precisely, here $L_\infty([0,T],L^p([0,1]))$ have the norm $\sup _{t\in[0,T]}\|f(t,\cdot)\|_{L^p}$ and $C([0,T],L^p[0,1])$ means an $ L^p([0,1])$-valued continuous function . Actually, I am confused to find the relation between these two types of solutions.
Since $[0,T]$ is compact, any continuous function on it must have a maximum, so the supremum $\sup_{t\in[0,T]} \lVert f(t,\cdot)\rVert_p$ is finite, and in fact is attained, when $f$ is continuous. (We're also using here that $\lVert\cdot\rVert_p$ is continuous.) In this way, we get a mapping $C([0,T],L^p([0,1])) \to L_\infty([0,T],L^p([0,1]))$. You didn't put a norm on $C([0,T],L^p([0,1]))$, but using the max as norm is fairly common with spaces of continuous functions; then the map $C([0,T],L^p([0,1])) \to L_\infty([0,T],L^p([0,1]))$ is an isometric embedding.
This seems to be the most obvious relationship between the two spaces. A good question to ask would be whether $L_\infty([0,T],L^p([0,1]))$ can be derived from $C([0,T],L^p([0,1]))$ in some way; perhaps it is the completion? (I haven't decided this for myself yet.)