I am studying the heat equation \begin{align*} u_t - \Delta u = f \end{align*}
where $u \in C^\infty(\bar{\Omega} \times (0,1])$ has compact support on $\Omega$ for all $t > 0$. My objective is to demonstrate that there exists a constant $C$ such that \begin{align*} \int_0^1 \int_\Omega |D^2u|^2 \ dx \leq C \int_\Omega |f|^2 \ dx \end{align*}
I have come up with the following derivation, but I am unsure of whether it is correct or not, so please correct me if I have (spectacularly) erred.
Using integration by parts one can find that \begin{align*} \int_\Omega |D^2 u|^2 \ dx &= \sum_{i,j = 1}^n \int_\Omega u_{x_ix_j}u_{x_ix_j} \ dx\\ &= \int_{\partial \Omega} \sum_{i,j = 1}^n u_{x_j}(u_{x_ix_j}\nu^i - u_{x_ix_i} \nu^j) \ dS + \int_{\Omega} |\Delta u|^2 \ dx\\ &= K_1 + \int_{\Omega} |\Delta u|^2 \ dx \end{align*}
where $K_1 < \infty$ is equal to the first integral and is finite since $u$ has compact support. Now, after substituting in the heat equation and applying a combination of Cauchy's inequality and Young's inequality, it can be found that \begin{align*} \int_\Omega |D^2 u|^2 \ dx &= K_1 + \int_\Omega |u_t - f|^2 \ dx\\ &\leq K_1 + \int_\Omega |u_t|^2 + 2|u_t||f| + |f|^2 \ dx\\ &\leq K_1 + C_1\int_\Omega |u_t|^2 + |f|^2 \ dx\\ &\leq K_2 + \int_\Omega |f|^2 \ dx \end{align*}
Therefore, \begin{align*} \int_0^1 \int_\Omega |D^2 u|^2 \ dxdd &\leq C\int_\Omega |f|^2 \ dx \end{align*}
since again $u$ has compact support and due to the Archimedean property of the real numbers.
Although cumbersome, I believe the above derivation is sound, but I'm not confident in it.