This is related to [these posts][1] and [here][2].

Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$.  Denote the accelerated $\ell^1$-direct sum Banach space 
$$
\bigoplus_{n \in \mathbb{N}}^{\ell^1} L^1([0,n])
:= 
\left\{
f \in L^1([0,\infty)):\,
\sum_{n=1}^{\infty} \left(
2^n\int_{x \in [n,n+1] } |f(x)| dx
\right) <\infty
\right\}.
$$

Alternatively, equip $L^1_{\mathrm{comp}}$ be the space of essentially compactly-supported $L^p$ functions, with their usual inductive limit topology obtained by the inductive system
$$
\left\{f \in L^1_{\mathrm{loc}}: \, 
\operatorname{esssupp}(f)\subseteq [0,n]
\right\}
\to
\left\{f \in L^1_{\mathrm{loc}}: \, 
\operatorname{esssupp}(f)\subseteq [0,m]
\right\}\qquad n\leq m;\, n,m \in \mathbb{N}
.
$$  Is the topology of $\oplus^{\ell^1}_{n}L^1([0,n])$ coarser than that of $L^p_{\mathrm{comp}}$ on their intersection?

It seems to me to be the case since $\oplus^{\ell^1}_{n}L^1([0,n])$ is much smaller than $L^p_{\mathrm{comp}}$.  


  [1]: https://mathoverflow.net/questions/358663/use-of-this-space-of-very-rapidly-decreasing-continuous-functions
  [2]: https://mathoverflow.net/questions/347318/can-l1-loc-be-represented-as-colimit