Discrete-time, continuous space heat equation inequality

Question

For $u\left(x, t_{n+1}\right)$ that solves \begin{equation} \frac{u\left(x, t_{n+1}\right)-u\left(x, t_{n}\right)}{\Delta t}=\Delta u\left(x, t_{n+1}\right) \end{equation} with $u\left(x, t_{n+1}\right)=0$ on $\partial \Omega$. I'm trying to show that \begin{equation} \int_{\Omega}\left|\nabla u\left(x, t_{n+1}\right)\right|^{2} \mathrm{~d} x \leq \int_{\Omega}\left|\nabla u\left(x, t_{n}\right)\right|^{2} \mathrm{~d} x \end{equation} My thought was to use the energy method by integrating after multiplying by $u(x,t_{t+1})$: \begin{equation} \int_\Omega \frac{u\left(x, t_{n+1}\right)-u\left(x, t_{n}\right)}{\Delta t} u(x,t_{t+1})= \int_\Omega \Delta u\left(x, t_{n+1}\right) u(x,t_{t+1}) \end{equation} Simplifying the left-hand side and using integration by parts and the boundary condition $u\left(x, t_{n+1}\right) = 0$ on $\partial \Omega$ on the right: \begin{equation} \frac{1}{\Delta t} \left( \int_{\Omega} u\left(x, t_{n+1}\right)^2 \mathrm{d} x - \int_{\Omega} u\left(x, t_{n+1}\right) u\left(x, t_{n}\right) \mathrm{d} x \right) = -\int_{\Omega} \left|\nabla u\left(x, t_{n+1}\right)\right|^2 \mathrm{d} x \end{equation} Rearrange to isolate the gradient term: \begin{equation} \int_{\Omega} \left|\nabla u\left(x, t_{n+1}\right)\right|^2 \mathrm{d} x = -\frac{1}{\Delta t} \left( \int_{\Omega} u\left(x, t_{n+1}\right)^2 \mathrm{d} x - \int_{\Omega} u\left(x, t_{n+1}\right) u\left(x, t_{n}\right) \mathrm{d} x \right) \end{equation} The right-hand side can be rewritten using the Cauchy-Schwarz inequality: \begin{equation} \int_{\Omega} \left|\nabla u\left(x, t_{n+1}\right) \right|^2 \mathrm{d} x \leq -\frac{1}{\Delta t} \left( \int_{\Omega} u\left(x, t_{n+1}\right)^2 \mathrm{d} x - \left( \int_{\Omega} u\left(x, t_{n+1}\right)^2 \mathrm{d} x \right)^{1/2} \left( \int_{\Omega} u\left(x, t_{n}\right)^2 \mathrm{d} x \right)^{1/2} \right) \end{equation} I've tried factoring out the common integral on the right-hand side and using the Poincare inequality, but it's not clear on how to proceed. What's the rest of the way to derive the desired inequality?