Let $f(x,t)$ be defined on some space $X \times (0, T)$ and suppose for all time $t$, $f(t) \in L^p(X)$. I am trying to think of an example where $f \not\in L^q(0, T; L^p)$.
By definition we have $$\|f\|_{L^q(0, T; L^p)} = \Big(\int_0^T \|f\|_{L^p(X)}^q\Big)^{(1/q)}$$ and so $$\Big(\int_0^T \|f\|_{L^p(X)}^q d\mu(t) \Big)^{(1/q)} \leq \Big(\max_{t \in (0,t)} \|f(t)\|_{L^p(X)}^q \int_0^T d\mu(t)\Big)^{(1/q)} \\= \mu(0,T)^{(1/q)} \max_{t \in (0,t)} \|f(t)\|_{L^p(X)}^q$$ and since $(0,T)$ has finite measure and $f(t) \in L^p(X)$ for all $t$, then is finite but this seems like a strange and incorrect result since it would imply if the if $f \in L^p$ then it is in $L^q(0, T; L^p)$ for all $q > 0$.
Is there an example of a function that is in $L^p$ for all $1 \leq p \leq \infty$ but not in $L^q(0, T; L^p)$?