Consider the following evolution equation

$$u_t=\Delta u$$ in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous Dirichlet boundary conditions.

It is known that this equation has a smooth global solution $u$. By differentiating $E(t):=\int_\Omega u^2 dx$, using integration parts and Poincaré's inequality we can prove that

$$|u(t)|_{L^2(\Omega)}\leq e^{ -\frac{1}{c^2}t}|u(0)|_{L^2(\Omega)}$$ where $c$ is the Poincaré constant. This means that the solution decays exponentially in $L^2(\Omega)$.

**My question is**: Does this decay result holds in $L^p(\Omega)$ for $p>2$ or maybe even $L^\infty(\Omega)$?
The fact that $p=2$ is very important here because after the integration by parts we get the term $\int_\Omega |\nabla u|^2dx$ which we know how to estimate in terms of $\int_\Omega u^2 dx$ via Poincaré's inequality.

ultracontractivity, introduced by Davies and Simon (I think here). The $L^\infty$ norm of $u(t + 1)$ is bounded by a constant times the $L^2$ norm of $u(t)$, which we already know decays exponentially with $t$. $\endgroup$ – Mateusz Kwaśnicki Jul 29 at 20:58