Spacial regularity of heat equation with zero initial condition

Question

Consider the heat equation with zero Dirichlet boundary condition and zero initial data in time interval $[0,T]$: $$ \partial_t u-\Delta u=f\;\;\text{in $\Omega$};\quad u(0)=0;\quad u|_{\partial \Omega}=0 $$ If we assume that $f\in L^2H^2(\Omega)$ (where $L^2H^2(\Omega)$ is an abbreviation of the Bochner space norm $L^2(0,T;H^2(\Omega)$) and $f(t)\in H^1_0(\Omega)$ for almost every $t$, can we conclude the following estimate? $$ \|\partial_t u\|_{L^2H^2(\Omega)}\leq C\|f\|_{L^2H^2(\Omega)}. $$ My idea is to intuitively take operator $\Delta$ on the both side of the original heat equation, thus consider the solution $v$ of the following heat equation $$ \partial_t v-\Delta v=\Delta f\;\;\text{in $\Omega$};\quad v(0)=0;\quad v|_{\partial \Omega}=0 $$ then let $u=(\Delta)^{-1} v$, $u$ seems to be the solution of the original heat equation and satisfies the estimate above. But it seems something went wrong here, could you help with me with finding the mistake?