Let $n\geq1$ and $\Omega\subset\mathbb{R}^{n}$ is expected to be a bounded Lipschitz domain with boundary $\Gamma$. $Q:=(0,T)\times \Omega$, $\Sigma:=(0,T)\times \Gamma$. $T>0$ is fixed and finite.
I have the following question:
Does there exists a result, is there a counterexample, or under which assumptions exists a result for an estimate of the kind:
\begin{equation} \|y\|_{L^{\infty}(Q)}\leq c(\|f \|_{L^{2}(Q)} +\|g\|_{L^{2}(\Sigma)}+\| y_{0}\|_{C(\bar{\Omega})}), \end{equation} for a linear parabolic pde of type \begin{alignat}{2} y_t - \Delta y+\alpha y&= f&\qquad &Q\\ \partial_{\nu}y+\beta y&=g& &\Sigma\\ y(0)&=y_0 & &\Omega, \end{alignat} for $y_{0}\in C(\bar{\Omega})$ or especially $y_{0}=0$, $\alpha\in L^{\infty}(Q)$, $\beta\in L^{\infty}(\Sigma)$, $\alpha, \beta \geq 0$ and further $f\in L^{\infty}(Q)$ and $g\in L^{\infty}(\Sigma)$.
More exactly, the question is, if we get this estimate also for dimension $n=3$ or higher with the right-hand side still in $L^{2}$-norm?
I know, that there exist results of this kind for the semilinear equation with similar assumptions, but for the right-hand side with $\|f \|_{L^{r}(Q)}$, $\|g \|_{L^{s}(\Sigma)}$ ($r>n/2+1$, $s>n+1$) instead of $L^{2}$, as written above.
Thank you!