Let $u$ be a solution to the heat equation in a domain $U \times [0,T]$. Let $f$ be a $C^2$ function on the closure of $U \times [0,T]$. Assume that $$f = |\nabla f|=0 \text{ on } \partial U \times [0,T].$$ Prove that
$$\int_{U_T} u(f_t + \Delta f) \text{dx dt} = \int_U u(x,T)f(x,T)-u(x,0)f(x,0) \text{ dx}.$$
From what I can gather regarding calculus, we need to show that the derivative with respect to time of $u(f_t + \Delta f)$ is equal to $u(x,t)f(x,t)$. This also means that $\Delta u(f_t + \Delta f)$ is also equal to $u(x,t)f(x,t)$. I am also sure that there is something special about the function $f$ because of its condition on the boundary. The problem is that I don't know what to do with the function inside. There is something I am missing with regards to that. Can someone help me?