The weighted Yamabe problem with boundary

We introduce a Yamabe-type flow \begin{align*} \left{ \begin{array}{ll} \frac{\partial g}{\partial t} &=(r^m_φ-R^m_φ)g \ \frac{\partial φ}{\partial t} &=\frac{m}{2}(R^m_φ-r^m_φ) \end{array} \right. ~~\mbox{ in }M \mbox{ and } H^m_φ=0 ~~\mbox{ on }\partial M \end{align*} on a smooth metric measure space with boundary $(M,g,v^{m}d{V}_g,v^{m}d{A}_g,m)$, where $R_{\phi }^m$ is the associated weighted scalar curvature, $r_{\phi }^m$ is the average of the weighted scalar curvature, and $H_{\phi }^m$ is the weighted mean curvature. We prove the long-time existence and convergence of this flow.