Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.

It is known that $S(t)$ doesn't induce a $C_0$-semigroup in $L^\infty(\Omega)$ because of the lack of continuity of $t \mapsto S(t)x,\; x\in L^\infty(\Omega)$ w.r.t the $L^\infty$-norm.

So my question is the following: is the integral $\int_0^T \|S(t)x\|_{L^\infty(\Omega)} dt$ well defined for some class of $x\in L^2(\Omega)$ and for every $T>0$?

Thanks!