Regularity on the boundary for the heat equation with linear source

Question

This is probably a known problem but I was not able to find exactly what I am looking for.

I have the following linear heat equation with zero-flux boundary conditions:

\begin{equation} \begin{cases} \dot{u} - \Delta u = u \quad \text{in} \quad \Omega;\\ \nabla u \cdot \boldsymbol{n} = 0 \quad \text{on} \quad \partial \Omega, \end{cases} \end{equation}

on a NON-CONVEX polygonal domain $\Omega \subset \mathbb{R}^2$, with $\Delta$ being the Laplace operator. From basic energy arguments, we have the following estimate

\begin{equation} \tag{1} \label{true_estimate} \|u(t)\|_{H^1(\Omega)} \leq C \|u(0)\|_{H^1(\Omega)} \exp(Ct), \qquad t >0. \end{equation}

Question: if $\Gamma \subset \partial \Omega$ is an edge of $\Omega$ -or a portion of an edge to avoid corner singularities-, do we have a similar estimate for the trace of $u$ on $\Gamma$? That is:

\begin{equation} \tag{2} \label{desired_estimate} \|Tr(u(t))\|_{H^1(\Gamma)} \leq C \left(\|Tr(u(0))\|_{H^1(\Gamma)} + \|u(0)\|_{H^1(\Omega)}\right)\exp(Ct), \qquad t>0. \end{equation}

If \eqref{desired_estimate} does not hold true, could you please provide a counterexample.

Challenges

• Estimate \eqref{desired_estimate} cannot be inferred from \eqref{true_estimate} via trace theorem because the trace of a $H^1(\Omega)$ function is only $H^{1/2}(\Gamma)$. So we try to exploit higher regularity in the interior.

• But we don't have enough regularity in the interior due to non-convexity. The best we can expect is a $H^{1+\varepsilon}(\Omega)$ estimate, with $0< \varepsilon < 1/2$. So, by the trace theorem we could get a $H^{1/2 + \varepsilon}(\Gamma)$ estimate on the boundary, with $0< \varepsilon < 1/2$.

Answer 1

The estimate $(2)$ is false even in a half-plane. Indeed, let $w$ be any solution to the heat equation on $\mathbb{R}^2 \times [0,\,\infty)$ that is even in $y$ (so $w$ solves the Neumann problem for the heat equation in the upper half-plane), and vanishes on the $x$-axis at $t = 0$. Then $w_R(x,\,y,\,t) := w(Rx,\,Ry,\,R^2t)$ solves the Neumann problem for the heat equation in the upper half-plane, vanishes on the boundary at $t = 0$, and has initial $H^1$ norm independent of $R$ (by the two-dimensionality of the spatial domain). We have in addition that $$f_R(t) := \int_{\mathbb{R}} |\partial_x w_R|^2(s,\,0,\,t)\,ds = Rf_1(R^2t).$$ Thus, if we let $u_R = e^tw_R$, then $u_R$ solves the desired Neumann problem in the upper half-plane, but at $t = R^{-2}$ the left side of $(2)$ is larger than $\sqrt{f_R(R^{-2})}\sim R^{1/2}$ and the right side is $\sim 1$ (independent of $R$). Taking $R \rightarrow \infty$ we see that such an estimate cannot hold.