Let $f \in C(\bar \Omega \times [0,T])$ and consider the following problem:
$\begin{align} \partial_tu -\Delta u&=f \quad \text{in} \; \Omega \times (0,T)\\ u(\cdot,0)&=u_0 \quad \text{in} \; \Omega \end{align}$
Assume that $u_0 \in L^2(\Omega)$. My question is: Can we deduce $u\in C^{1+a, \frac{1+a}{2}}(\Omega \times (0,T))$ for some $0<a<1$? Or can we at least obtain a Hoelder estimate only in terms of the space variable?
My thought was that since $f \in C(\bar \Omega \times [0,T])$ implies $f \in L^2(\Omega \times (0,T))$, we could apply the known $L^2-$regularity and obtain $u\in L^2(0,T;H^1_0(\Omega))\cap H^1(0,T;H^{-1}(\Omega)) \cap W^{2,1}_p(\Omega \times (\sigma,T))$ for all $1 \leq p <\infty\;$ and for any $\sigma>0$. Then, by a t-Sobolev embedding we could obtain the desired regularity.
Is the above correct or did I miss anything? I would appreciate any help or hints!
Thanks in advance!
The fact that $u \in W^{2,1}_p(\Omega \times (\sigma,T))$ does not follow from $L^2$-theory using energy estimates; you need to use the fact that $f$ lies in $L^p.$ Your idea is correct to show this and deduce the result using Sobolev embedding, but you would need to use a suitable $L^p$ estimates for the form $$ \lVert u \rVert_{W^{2,1}_p(\Omega \times (\sigma,\rho))} \leq C \left( \lVert f \rVert_{L^p(\Omega \times (0,T))} + \lVert u \rVert_{L^2(\Omega \times (0,T))} \right),$$ where $0 < \sigma < \rho < T.$ Note this uses a global estimate, which requires some regularity of $\partial\Omega,$ but you can work around this since the question only asks for interior regularity. Then you would use the embedding $$ W^{2,1}_p(\Omega \times (\sigma,\rho)) \hookrightarrow C^{1+\alpha,\frac{1+\alpha}2}(\overline \Omega \times [\sigma,\rho]) $$ for $p > n,$ where $\alpha = 1-\frac{p}n.$