The inequality is obviously false, because if $f\equiv c$ is a constant, then $u\equiv c$ remains constant. Since $\Omega$ is bounded, this $f$ is $L^2(\Omega)$, and yet $\|u(t)\|_\infty=|c|$ does not decay as $t\to+\infty$.
I suppose that you make a confusion with the heat equation in the hole space ($\Omega={\mathbb R}^N$).