hi im studying heat equation in open,bounded,regular $\Omega \subset R^{n}$ $$\partial_{t}u-\Delta u=0 \ (x,t)\in \Omega \times (0,\infty)$$ with $$u=0 \in \partial \Omega\times (0,\infty) and \ u_{0}\in L^{2}(\Omega)$$ where the solution is given by this formula: $$u(x,t)=\sum_{n\geq 1} a_{n}(t)e_{n}(x)$$ where $$a_{n}(t)=a_{0}\exp(-\lambda_{n}t) $$ and $\lambda_{n}$ are the eigenvalues of laplacian and i want to show that $u \in L^{2}((0,\infty),H_{0}^{1}(\Omega)\cap H^{2}(\Omega))$ so i consider that the space $H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ is equipped with norm: $$N(u)=\lvert u\rvert_{H_{0}^{1}(\Omega)}+\|\Delta u\|_{L^{2}}$$
and i take $t,s \in (0,\infty)$ and i show that :$N^{2}(u(t)-u(s))\to 0$ as $t\to s$ in fact $N^{2}(u)\leq C \lvert u\rvert_{H^{2}}^{2}$ which is by elliptic regularity $\leq M \lvert \Delta u\rvert_{L^{2}}^{2}$ but im not sure that this proof is true and if it is the case please i need a hint, thanks