Consider the heat equation
\begin{equation*} \begin{cases} u_{t} - \Delta u = f &\text{in }(0,T)\times\Omega\\ u(x,t) = g(x,t) &(t,x)\in [0,T]\times\partial\Omega\\ u(x,0) = h(x) &\text{on }\{0\}\times\Omega. \end{cases} \end{equation*}
When $g(t) = 0$, we have the estimate from Evans:
\begin{aligned} \text{ess sup}_{0 \leqslant t \leqslant T} \|u(t)\|_{H_{0}^{1}(\Omega)} + \|u\|_{L^{2}(0,T;H^{2}(\Omega))} + \|u'\|_{L^{2}(0,T;L^{2}(\Omega))} \leqslant C(\|f\|_{L^{2}(0,T;L^{2}(\Omega))} + \|h\|_{H^{1}_{0}(\Omega)}). \end{aligned}
I am curious about the case when $g\neq 0$ and wondering if anyone has a reference they can point me to. I know in the case of the Poisson equation, the boundary data would appear in the estimate taken in a trace space such $H^{3/2}(\partial\Omega)$. Would it be similar here?
Edit: I've added a sketch of my solution attempt. My idea was to try to leverage the existing estimate. Namely, consider $u_g \in L^{2}(0,T; H^{2}(\Omega))$ with $u_{g}' \in L^{2}(0,T; L^{2}(\Omega))$ which satisfies, for fixed $t$, $\mathfrak{T}u_g(\cdot,t) = g(\cdot,t)$, where $\mathfrak{T} : H^{2}(\Omega) \to H^{3/2}(\partial\Omega)$ is the trace operator. When $\Omega$ is Lipschitz I believe $\mathfrak{T}$ should have a continuous right inverse. Setting $\tilde{u} := u - u_g$, we have
\begin{equation*} \begin{cases} \tilde{u}_{t} - \Delta \tilde{u} = f - (u_g)_t + \Delta u_g &\text{in }(0,T)\times\Omega\\ \tilde{u}(x,t) = 0 &(t,x)\in [0,T]\times\partial\Omega\\ \tilde{u}(x,0) = h(x) &\text{on }\{0\}\times\Omega. \end{cases} \end{equation*}
if $u_g(x,0) = 0$ as well. Applying Evans' estimate gives
\begin{aligned} \text{ess sup}_{0 \leqslant t \leqslant T} \|\tilde{u}(t)\|_{H_{0}^{1}(\Omega)} + \|\tilde{u}\|_{L^{2}(0,T;H^{2}(\Omega))} + \|\tilde{u}'\|_{L^{2}(0,T;L^{2}(\Omega))} &\leqslant C(\|f\|_{L^{2}(0,T;L^{2}(\Omega))} + \|\Delta u_g\|_{L^{2}(0,T;L^{2}(\Omega))} + \|u_g'\|_{L^{2}(0,T;L^{2}(\Omega))} + \|h\|_{H^{1}_{0}(\Omega)})\\ &\leqslant C(\|f\|_{L^{2}(0,T;L^{2}(\Omega))} + \|u_g\|_{L^{2}(0,T; H^{2}(\Omega))} + \|u_g'\|_{L^{2}(0,T;L^{2}(\Omega))} + \|h\|_{H^{1}_{0}(\Omega)})\\ &\leqslant C(\|f\|_{L^{2}(0,T;L^{2}(\Omega))} + \|g\|_{L^{2}(0,T; H^{3/2}(\partial\Omega))} + \|u_g'\|_{L^{2}(0,T;L^{2}(\Omega))} + \|h\|_{H^{1}_{0}(\Omega)}). \end{aligned}
I thought I could use this estimate along with a triangle inequality to obtain the estimates on $u$, but the term $\|u_g'\|_{L^{2}(0,T; L^{2}(\Omega))}$ is giving me some pause. Maybe I could define it so that $u_g'(t) = 0$ in a region away from $t=0$ and the boundaries and then have to vary to 0 in a way that still preserves $u_g(\cdot,t) \in H^{2}(\Omega)$, but I'm having trouble with this part and what the estimate would look like.
Final edit: For anyone who comes across this question in the future, I found two very useful texts that discuss spaces for the boundary data and trace theorems for the space $H^{s}(0,T; H^{2s}(\Omega))$. These are
Optimal Lp-Lq estimates for parabolic boundary value problems with inhomogeneous data (Denk, Hieber, and Prüss)
Nonhomogeneous boundary value problems and applications (Lions and Magene)