What is the definition of $L^p(0,T; L^p_{loc}(R^n)), p\ge 1$?

Question

I saw the space $L^p(0,T; L^p_{loc}(\mathbb R^n))$ in a paper about parabolic equation. But I only know $$ L^p_{loc}(\mathbb R^n) =\{ u:\mathbb R^n\rightarrow \mathbb R: u\in L^p(V), \forall ~ V\subset \subset \mathbb R^n \} $$ But I don't know what is $L^p(0,T; L^p_{loc}(\mathbb R^n))$ , and what is its norm.

Besides, I want a book can be used to find the notation of parabolic equation.

Answer 1

They are Bochner Spaces. Basically functions $f(t,\cdot)$ which are in $L^p_{loc}(\mathbb{R}^n)$ independent of time parameter $t \in [0,T]$. The norm of $L^p([0,T],L^p(\mathbb{R}^n))$ is $$\|f\|^p_{L^p([0,T],L^p(\mathbb{R}^n))}:=\int_0^T ||f(t,\cdot)||^p_{L^p(\mathbb{R}^n)}dt$$.

Some references

• https://en.wikipedia.org/wiki/Bochner_space.
• H.Brezis, Functional Analysis,Sobolev Spaces and Partial Differential Equations, Chapter 10.
• Any book about time evolving, Parabolic, or Hyperbolic PDE.