A pointwise bound for $\frac{\partial}{\partial t}u=\Delta u+ au$

Question

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.