Given a differential operator $\mathrm{K}=\partial/\partial x$ and its domain $D(\mathrm{K})=\left\{\mathrm{f} \in \mathrm{L}^2(0,1) |\mathrm{f}(0)=0\right\}$. What I know is that this operator $\mathrm{K}$ cannot generate a $C^0$-semigroup and if I change the boundary condition to $\mathrm{f}(1)=0$, then it can generate a $C^0$-semigroup. I think it will be related to some singularity at 0 point. But I don't know how to show it mathematically. Can someone give a specific example or intuitive explanation why this is the case?
2026-04-13 23:46:08.1776123968
Question on differential(shift) semigroup
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