This is a [cross-post from Computational Science](https://scicomp.stackexchange.com/questions/20074/bounded-input-boundaed-output-stability-for-heat-equation-proof-or-counter-exam "link").


I am interested in proving or obtaining a counterexample to the following conjecture.


Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let $u\_d\in H^{1/2}(\partial\Omega) \times \mathbb{R}^+$. Suppose $u\in L^2((0,\infty); H^1(\Omega))$ is the weak solution of 
$$
\left\{
\begin{align\*}
 u\_t- \nabla \cdot \nabla u &= 0&\, \mathrm{in}\,& (\Omega\times \mathbb{R}^+) \\
 u &= u\_d&\, \mathrm{on}\,& (\partial\Omega \times \mathbb{R}^+)\\
 u &= 0&\, \mathrm{at}&\, (\Omega \times \{0\})
\end{align\*}
\right.
$$
Then there exists a constant $C$ dependent only on the domain $\Omega$ such that
$
\left\| u \right\|\_{L^\infty((0,\infty); L^2(\Omega))} \le C \left\| u\_d \right\|\_{L^{\infty}((0,\infty); H^{1/2}(\partial\Omega))}.
$


I haven't been able to find many texts that treat the heat equation with time varying boundary conditions. Also, feel free to change the Sobolev spaces if you need to.


Addendum #1 Response to Denis Serre. Thanks for the response. I would like to check my understanding. To simplify things I am going to consider a 2-dimensional problem.


We modify the problem as follows.


Let $\Omega \subset \mathbb{R}^2$ be defined as $\Omega = \{(x,y)\in \mathbb{R}^2: x>0\}$ 


Suppose $u\in L^2(\mathbb{R}^+; H^1(\Omega)) $ is the weak solution of the above initial boundary value problem.


Step #1, Laplace Transform in $t$.


$0=\tau U(\tau,x,\eta) - \frac{\mathrm{d}^2}{\mathrm{d}y^2} U(\tau,x,\eta) - \frac{\mathrm{d}^2}{\mathrm{d}x^2}U(t,x,\eta)$


Step #2, Fourier transform in $y$.


$0 = \tau\, \mathcal{U}(\tau, x,\eta) + |\eta|^2 \mathcal{U}(\tau, x,\eta) - \frac{\mathrm{d}^2}{\mathrm{d}x^2} \mathcal{U}(\tau,x,\eta)$


Step #3


Solve the resulting ODE.


$\mathcal{U}(\tau, x,\eta) = \mathcal{U}\_d(\tau, \eta) \exp(-x\sqrt{\tau+|\eta|^2)}$


Step #4


Estimate $L^2$ norm.


$\int\_{x\in \mathbb{R}^+}|\mathcal{U}|^2\, \mathrm{d}x = \cdots = |\mathcal{U}\_d(\tau, \eta)|^2 \frac{1}{2\sqrt{\tau+|\eta|^2}} $


Step #5


Go back by Plancherel


$$
\begin{align\*}
\int\_{(x,y)\in \Omega} |U(\tau, x,y)|^2 \mathrm{d}x \mathrm{d}y 
&= \int\_{\eta\in \mathbb{R}} \int\_{x\in \mathbb{R}^+} |\mathcal{U} (\tau,x,\eta)|^2 \mathrm{d}x\, \mathrm{d}\eta \\
&= \int\_{\eta\in \mathbb{R}} |\mathcal{U}\_d(\tau, \eta)|^2 \frac{1}{2\sqrt{\tau+|\eta|^2}}\mathrm{d}\eta\\
&= \int\_{\eta\in \mathbb{R}} |\mathcal{U}\_d(\tau,\eta)|^2 \frac{1}{2 \sqrt{\tau+ |\eta|^2}} \frac{(1+|\eta|^2)^{1/4}}{(1+|\eta|^2)^{1/4}}\, \mathrm{d}\eta \\
&\le \frac{1}{2\sqrt{\tau}} \left\|U\_d(\tau,\cdot)\right\|^2\_{H^{1/2}(\partial\Omega)}
\end{align\*}
$$


I honestly don't know how to apply Paley-Wiener here. Any hints would be appreciated. If I am totally off track, please let me know too. Thanks.



Bounded input Bounded output stability for heat equation
The theory of linear Intial-boundary value problem proceeds the following way. In your case, because of the zero initial condition, you may replace ${\mathbb R}^+$ by $\mathbb R$ and define $u\equiv0$, $u\_d\equiv0$ for $t<0$. Then what matters is the special case where $\Omega$ is a half-space, say $\{x\_d>0\}$. In this simpler situation, you can (and you must) perform a Fourier transform in the tangential variables $x\_1,\ldots,x\_{d-1}$, and a Laplace transform in time. You end up with an ODE, parametrized by the frequency variables $\eta\in{\mathbb R}^{d-1}$ and $\Re\tau>0$ :
$$\tau v+|\eta|^2v-\frac{d^2v}{dx\_d^2}=0,\qquad v:=\hat u$$
with an ''initial'' data
$$v(\tau,\eta,0)=\hat u\_d(\tau,\eta).$$
Here, $x\_d\mapsto v$ is the unique square integrable solution over ${\mathbb R}^+$. 


Write the explicit form of $v(\tau,\eta,\cdot)$, estimate its $L^2$-norm (or every $H^s$-norm you wish). Then go back by Plancherel and Paley-Wiener. You obtain an accurate estimate in weighted spaces $L^2(e^{-\gamma t}dt;L^2(\Omega))$.


The method is detailed in my book with S. Benzoni-Gavage, in the case of hyperbolic equations (like the wave equation), but it can be applied to any well-posed IBVP. It is also used to detect the well- or ill-posedness (key word : Lopatinskii condition).


