Upgrading the regularity of a solution in a Bochner space

Question

Suppose $\Omega \subset \mathbb{R}$ is bounded and that

$$u \in C(0,T; L^{2}(\Omega)) \cap L^{2}(0,T; H^{1}(\Omega))$$

We are in 1D so the Sobolev embedding $H^{1}(\Omega) \hookrightarrow C(\Omega)$ applies.

Question: Is there any way to conclude from this information that $u \in C(\Omega \times [0,T])$?

Answer 1

I think that $f(t,x) = t^x$ with $T = 1$ and $\Omega = [0,1]$ could be a counterexample:

The function $t \mapsto f(t, \cdot) \in L^2(\Omega)$ should be continuous due to the dominated convergence theorem. Thus, $f \in C(0,T;L^2(\Omega))$.

On the other hand, $\partial_x f(t,x) = t^x \log(t)$ is square-integrable, i.e., $f \in L^2(0,T;H^1(\Omega))$.