I'm working on the mathematical theory of parabolic equations. The prototype of such equations is heat equation given as follows : Let $\Omega$ be a bounded region of the space and $T>0$ a fixed time. In $\Omega_T=(0,T)\times \Omega$ we consider the following equation $$u_t =\alpha\Delta u -a(x)u,$$ $$u(0,x)=f(x),$$ where $f$ is the initial condition, $a$ a bounded potential, $\alpha>0$ is a constant, and $\Delta$ is the Laplacian. I'm wondering to know the physical meaning of the coefficient $a$ (and may be $\alpha$) and it's role in the heat process? Any reference or suggestion would be helpful.