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$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$ \begin{cases} \partial_t u-\Delta u=0 \quad(x,t)\in \Omega\times (0,T]\\ u(x,0)=u_0(x)\\ u(x,t)=0 \quad (x,t)\in \partial\Omega\times (0,T] \end{cases} Whether there is estimation like: $$ \|\nabla u(\cdot,t)\|_{L^\infty(\Omega)}\leq C\|\nabla u_0\|_{L^\infty(\Omega)}, $$ here $C$ is a constant independent of $u$ and $t$.

In fact, I wish the following estimate holds: If $u_0\in C(\bar\Omega)\cap H^1_0(\Omega)$, and satisfies that there exists $d_0>0$ and $C_0>0$ such that if $x\in \Omega$ with ${\rm dist}(x,\partial\Omega)\leq d_0$ then $$ |u_0(x)|\leq C_0{\rm dist}(x,\partial\Omega) $$ Then the solution $u(\cdot, t)$ of the heat equation will still satisfy that if $x\in \Omega$ with ${\rm dist}(x,\partial\Omega)\leq \kappa d_0$ then $$ |u(t,x)|\leq \lambda C_0{\rm dist}(x,\partial\Omega) $$ where $\kappa$ and $\lambda$ are constants only dependent on $\Omega$.

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    $\begingroup$ Your assumption is equivalent to saying that $|u_0(x)| \leq C\phi(x)$, where $\phi$ is the first eigenfuntion. Then $|u(t,x)| \leq Ce^{-\lambda t}\phi(x)$, with $\lambda>0$ the first eigenvalue. For the boudary nehaviour of $\phi$ see for example Lemma 4.6.1, Heat kernels and spectral theory by Davies. $\endgroup$
    – Giorgio Metafune
    Commented Mar 11, 2022 at 10:01
  • $\begingroup$ @GiorgioMetafune Thanks for your answer so much! It has been really helpful for me. Could you explain moreover, why my assumption is equivalent to $|u_0(x)|\leq C\phi(x)$, since here I only assume $|u_0(x)|\leq C{\rm dist}(x,\partial\Omega)$ when $x$ is sufficiently near the boundary. $\endgroup$
    – Tibeku
    Commented Mar 11, 2022 at 12:47
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    $\begingroup$ Yes, sure. The first eigenfunction is positive in the interior, so any estimate near the boundary extends to a global estimate, enlarging the constant $C$. $\endgroup$
    – Giorgio Metafune
    Commented Mar 11, 2022 at 13:06
  • $\begingroup$ @GiorgioMetafune Thank you so much for your help! May I ask one more question? Lemma 4.6.1 in book Heat kernels and spectral theory requires the domain to be $C^2$ and Lemma 4.6.1 does not hold in non-smooth domain. Do you think in convex polyhedron domain case there still holds: if $|u_0(x)|\leq C_0d(x)$, then $|u(t,x)|\leq Cd(x)$ where constant $C$ independent of $t$? $\endgroup$
    – Tibeku
    Commented Mar 11, 2022 at 13:37
  • $\begingroup$ This should be true. If $\Omega$ is convex, then $\Delta d$ is concave (see mathoverflow.net/questions/291308/…). If $\Omega$ is also smooth, this implies $\Delta d \leq 0$ and then $d_t-\Delta d \geq 0$. The maximum principle for the heat equation gives $|u(t,x)| \leq Cd(x)$. This estimate does not depend on smoothness if $\Omega$ is convex and should give the result by some approximation, I guess. $\endgroup$
    – Giorgio Metafune
    Commented Mar 11, 2022 at 14:56

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