Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$

Question

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please correct it .

How to show the existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$ ?

Answer 1

Let $\tau=t_0-t$, $u=e^{-f}$,then ,$\partial_\tau u=\partial_t f u=(-\Delta f+|\nabla f|^2-R)u=\Delta u -Ru$.So,we get :

$\partial_\tau u=\Delta u-Ru$

It is a standard linear parabolic equation.