EDIT2: I believe the estimate required is false. There is some evidence that I have added to this post. I only believed it was true because it seemed like it was used in certain papers. However, in those papers the implied statement was that if $|f(t)|\leq e^{Ct}$ then $\|u(t)\|_{L^2}\leq e^{Ct}$ (with the same constant $C$). In this case, it is indeed true, as can be confirmed using standard energy estimates.
Consider the heat equation with an inhomogeneous (Dirichlet, Neumann or mixed) boundary condition in the half line. Can we estimate the solution $u$ in terms of $f$ in a time-independent way? More precisely:
Let $f\in C^{\infty}([0,\infty]),a,b\geq 0, (a,b)\neq (0,0)$, satisfying the compatibility conditions $f^{(j)}(0)=0$ for all $j\in \mathbb{N}$, and $u(t,x)$ being the 'reasonable' solution of $$\begin{cases}\partial_tu(t,x)=\partial_{xx}u(t,x)& (t,x)\in(0,\infty)\times \mathbb{R}_+\\ a\partial_x u(t,0)-bu(t,0)=f(t) & t>0\\ u(0,x)=0 & x>0. \end{cases}$$ Then we have an estimate such as $$ \|u\|_{L^p(0,t;L^q(\mathbb{R}_+))}\leq C_{a,b}\|f\|_{L^r(0,t)}\qquad \forall t\geq 0, $$ for some $p,q,r\geq 1$ (not sure which would work, so anything goes).
I assumed it should work with Lebesgue norms, but if it does with some more exotic norms that's also fine as long as they don't depend explicitly on $t$.
I know the explicit formula for the solution using the Heat kernel but I still can't derive any estimate using Sobolev norms. The main issue is the constant $C$ being independent from $t$.
Example. If we choose $f=1$, $(a,b)=(1,0)$ then the solution is $$u(t,x)=\int_0^t\frac{-2}{\sqrt{4 \pi s}}e^{-\frac{x^2}{4s}}\,\mathrm{d}s \implies u(t,x)=-x\left(\frac{2e^{-\frac{x^2}{4t}}\sqrt{t}}{\sqrt{\pi}x}- \text{Erfc}\left(\frac{x}{2\sqrt{t}}\right)\right), $$ so that for all $k\in \mathbb{N}$, $$\|u(t)\|_{L^2(\mathbb{R}_+)}= Ct^{3/4}\not \leq C'\sqrt{t}= C'\|f\|_{H^k(0,t)}. $$ This denies the estimate for $\|u\|_{L^{\infty}(0,t;L^2)}$ and $\|f\|_{H^k(0,t\times \mathbb{R}_+)}$ for any $k\in \mathbb{N}$. However, $$\|u(t)\|_{L^{\infty}(\mathbb{R}_+)}= \frac{\sqrt{t}}{\sqrt{\pi}}\leq C\|f\|_{L^2(0,t)}.$$ Also this $f$ does not satisfy the compatibility conditions, so not sure if it's even a valid example.
EDIT: Simplified the question
EDIT: I gathered some evidence that the estimate is not possible even when $a=0$...
Suppose $a=0$, and that the inequality holds for some (a priori) time dependent constant $C(t_0)$. For each solution of the Heat equation apply the rescaling $u(t,y)\mapsto u_{t_0}(t,y):=u(t_0t,\sqrt{t}_0y)$. Now $u_{t_0}$ solves the same heat equation (after replacing $f$ with $f_{t_0}(t)=f(t_0t)$) with $t_0=1$. Therefore we have $$\|u_{t_0}\|_{L^p(0,1;L^q)}\leq C(1)\|f_{t_0}\|_{L^r(0,1)}$$ Which is equivalent to $$t^{-1/p-1/(2q)}\|u\|_{L^p(0,t_0;L^q)}\leq C(1)t_0^{-1/r}\|f\|_{L^r(0,t_0)} $$ So for the constant to be time-independent we need $$\frac{1}{r}=\frac{1}{p}+\frac{1}{2q}$$ Let $K(t,y)$ be the heat kernel. We have $$\|u\|_{L^p(0,t)}\leq\|-2\partial_y K\|_{L^{s}(0,t)}\|f\|_{L^r(0,t)} $$ where $s$ is given by $$\frac{1}{s}=1+\frac{1}{p}-\frac{1}{r}=1-\frac{1}{2q}. $$ Now we ask ourselves for which $s,r$ does $\|\partial_yK\|_{L^s(0,t;L^q(\mathbb{R}_+))}$ converge? The answer is whenever $$\frac{1}{2q}+\frac{1}{s}>1. $$ The last two conditions are in contradiction with each other. Therefore, no time independent estimate is possible in the Dirichlet case.
