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 $E(t):=\int_\Omega u^2 dx$ and using integration parts we get $$E'(t)=2\int_\Omega uu_t=2\int_\Omega u\Delta u=-2\int_\Omega |\nabla u|^2.$$ Now by using Poincaré's inequality we get $$E'(t)\leq (-\frac{2}{c^2})E(t)$$ where $c$ is the Poincaré constant. This implies that $$|u(t)|_{L^2}\leq e^{\left( -\frac{1}{c^2}\right)t}|u(0)|_{L^2}$$ which means that the solution decays exponentially.
My question is: can we prove the above property but in $L^p(\Omega)$ for $p>2$ or maybe even $L^\infty(\Omega)$? $p=2$ is very important in the proof I gave because after the integration by parts we get the term $\int_\Omega |\nabla u|^2dx$ which we know how to estimate in terms of $\int_\Omega u^2 dx.$