I have a question regarding conditions on data $f$, $u_0$ to obtain the solution of the parabolic problem.
Renardy and Rogers, An Introduction to Partial Differential equations (2nd ed.) states
After Theorem 11.3 they add
Lemma 11.4. Suppose that $u\in L^2((0,T),V)\cap H^1((0,T),V^∗)$. Then, in fact, $u \in C([0,T],H)$.
Regarding this lemma, they say
This shows that Theorem 11.3 is optimal; i.e., if we want a solution with the regularity guaranteed by the theorem, then the assumptions which we made on $f$ and $u_0$ are necessary.
Question: why does Lemma 11.4 imply the claimed necessity?
Thoughts: Suppose the solution satisfies $u\in L^2((0,T),V)\cup H^1((0,T),V^*)$. We want to show we must have $f\in L^2((0,T),V^*)$ and $u_0 \in H$.
Since $u\in H^1((0,T),V^*)$ we have $\frac{d u}{d t}\in L^2 ((0,T),V^*)$.
Since $u\in L^2((0,T),V)$, we have $u(t)\in V$ for a.e. $t$, and thus $A(t)u(t)\in V^*$ for a.e. $t$. From the continuity of $A(t)$ we have $A(\cdot)u(\cdot)\in L^2 ((0,T),V^*)$.
These two above force $f$ to be in $ L^2((0,T),V^*)$. I have not used Lemma 11.4.
Lemma 11.4 implies $u(0)\in H$, but $u(0)=u_0$ and thus we must have $u_0\in H$.
But it sounds a bit tautological since $u(0)=u_0\in H$ is meant to be given anyway, and it feels I needed to use Lemma 11.4 to claim $f\in L^2((0,T),V^*)$.