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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!

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  • $\begingroup$ Many thanks to Prof. Alex for the editing. $\endgroup$ – Rabat Mar 16 at 20:52
  • $\begingroup$ The integral is indeed well-defined. Firste, $|S(t) x|$ is bounded (pointwise) by the convolution of the Gauss–Weierstrass kernel $k_t$ and $|x|$, and by Cauchy–Schwarz, this convolution is bounded by $\|k_t\|_2 \|x\|_2 = c t^{-1/2} \|x\|_2$. Second, $S(t) x$ is jointly continuous as a function on $(0, \infty) \times \Omega$, and thus $\|S(t) x\|_\infty$ is continuous on $(0, \infty)$. $\endgroup$ – Mateusz Kwa?nicki Mar 16 at 21:27
  • $\begingroup$ I leave the above as a comment, as I believe this question is more suitable for Math.SE. $\endgroup$ – Mateusz Kwa?nicki Mar 16 at 21:28
  • $\begingroup$ Thank you for these useful information. Could you please indicate me any reference for that, especially for the continuity. $\endgroup$ – Rabat Mar 16 at 22:23
  • $\begingroup$ Joint continuity of $S(t) x$ is a consequence of the dominated convergence theorem and joint continuity of heat kernel. The latter property is classical, and I suppose it can be found in most textbooks on heat equation or parabolic PDEs (or Brownian motion); but I do not have a reference off the top of my head. $\endgroup$ – Mateusz Kwa?nicki Mar 16 at 23:28

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