How to estimate $\vert \vert u_t \vert\vert _{L^2(0,T;H^{-1}(\Omega))}$

Question

For heat equation $u_t-\Delta u=f$ with $u(x,0)=u_0,$ I have shown that $\vert \vert u \vert\vert ^2 _{L^2(0,T;H_0^{1}(\Omega))}\leq \vert \vert f \vert\vert ^2 _{L^2(0,T;H^{-1}(\Omega))} + \vert \vert u_0 \vert\vert ^2 _{L^2(0,T;H_0^{1}(\Omega))}.$ But can't do the estimate $\vert \vert u_t \vert\vert _{L^2(0,T;H^{-1}(\Omega))}\leq C(\vert \vert f \vert\vert _{L^2(0,T;H^{-1}(\Omega))} + \vert \vert u_0 \vert\vert _{L^2(0,T;H_0^{1}(\Omega))})$. How do I estimate?