I am considering a Cauchy problem for linear reaction-diffusion PDE of the form: $$\dfrac{\partial u}{\partial t} = a\Delta u + b(x)u, \quad u(t=0,x) = u_0(x), \quad x\in\mathbb{R}^{n},$$ where $\Delta$ is the Euclidean Laplacian operator, the constant $a>0$, and $b(x)$ is a given smooth function. Is there a standard approach/reference for solving this class of problems?
I know that if $b$ were constant, then we could do the change of variable $v := e^{-bt}u$, and obtain a Cauchy problem with heat PDE: $\frac{\partial v}{\partial t} = a\Delta v$, $v(t=0,x)=u_0$. Are there similar transformations known for the case $b$ depends on $x$ but not on $t$?