4

In this question I asked whether the Dirichlet PDE: $$\frac{\partial}{\partial t}u=\Delta u+ au$$ over a bounded smooth open subset $\Omega \subset \mathbb{R}^N$ has a bound in the form $$|u(t)|_{L^2(\Omega)}\leq Me^{\omega t}.$$ I wonder if we have a pointwise version of such an estimate. That is: $$|u(t,x)|\leq M_x e^{\omega_xt}$$ where $M_x$ and $\omega_x$ are constants that may depend on $x$. I thought about using some comparison between $L^2$-norm and $L^\infty$-norm, but we only have $$|.|_{L^2}\leq C|.|_{\infty}.$$ I don't know but I feel that only finding a closed form for the solution will solve the problem.

user165633
  • 2,941
  • What spatial boundary conditions are you imposing (if any)? And what can you say about the boundary of the region? Are you saying that the boundary is a smooth manifold? – Disintegrating By Parts Oct 10 '17 at 23:24
  • 2
    This is true, and $\omega_x$ need not depend on $x$. The main point is that the an integral norm at time $t$ controls uniform norm at times $t'>t$, due to the smoothing effect of the heat equation. But making this precise takes work, so I'd start by looking up "parabolic maximum principle". –  Oct 10 '17 at 23:45
  • See https://math.stackexchange.com/questions/567566/prove-the-estimate-ux-t-le-ce-gamma-t?rq=1 – xpaul Oct 11 '17 at 17:35

0 Answers0