The solution of homogenous heat equation in bounded regular domain $\omega$ of $R^{n}$ is $$u(t,x)=\sum_{n\geq 1}a_{n}(0)\exp(-\lambda_{n}t)e_{n}=S(t) u_{0}$$ where $e_{n}$ is Hilbert basis of $L^{2}(\omega)$ and so $S(t)$ is a semigroup not strongly continous on $0$, I know that I have to find a sequence $u_{n}$ with norm equal to $1$ and $\epsilon \to 0$ and $t_{n}\to 0$ such that: $\|S(t_{n})u_{n}-u_{n}\|_{2}>\epsilon$ but I didn't find such sequence, so what i propose is to take the following sequence $u_{n}=\dfrac{u_{1}}{\|u_{1}\|^{{2}}}$ where $u_{1}=\sum_{n\geq 1} a_{n}(0) e_{n}$ (then $\|u_{1}(0)\|_{2}=\sum_{n\geq 1}\lvert a_{n}(0)\rvert^{2}$) so $\|u_{n}\|^{2}=1$ and we take $t_{n}=\dfrac{1}{\lambda_{n}}$then: $\|S(t_{n})u_{n}- u_{n}\|_{2}=\dfrac{1}{\|u_{1}\|^{{2}}}\left(\lvert \exp(-1)-1\rvert^{2}\right)\sum_{n\geq 1} \lvert a_{n}(0)\rvert^{2} =\lvert \exp(-1)-1\rvert^{2}$ But I think that my counter example is false?
Thank you