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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 $\int_\Omega u^2 dx.$

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  • $\begingroup$ I think you can, using Gronwalls inequality. $\endgroup$ – mattos Jul 29 '19 at 3:00
  • $\begingroup$ @Mattos If I take $E(t)=\int u^p$, then $E'(t)=pd\int u^{p-1}\Delta u+pa\int u^p=-pd\int \nabla u^{p-1}.\nabla u+paE(t)$. The problem is I don't know any kind of inequality between $\int \nabla u^{p-1}.\nabla u dx$ and $\int u^p dx$ except when $p=2$ which is Poincaré's inequality. $\endgroup$ – David Lingard Jul 29 '19 at 14:55

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