If we have the heat equation on a simply connected (and smooth) domain $\Omega$ with boundary $\Gamma$ with Robin boundary conditions, so that: $$ \epsilon u_t - \Delta u = -1 \quad on \quad \Omega \\ \nabla u \cdot n + u = Q \quad on \quad \Gamma \\ u(0) = u_0 \quad on \quad \Omega \cup \Gamma. $$ Then can we get an $L^p (0, T; L^2(\Omega))$ bound on $u$ that does not depend on $1/ \epsilon$? (Here $p > 0$, and maybe $\infty$). An $L^p (0, T; L^2(\Gamma))$ bound would be just as good.
If I follow usual energy techniques, then I can use Gronwall's inequality to give a bound, but it depends on $1 / \epsilon$. I don't want it to depend on $\epsilon$, as I will later be taking $\epsilon \rightarrow 0$.
Thanks!