How to transform parabolic equation into heat equation?

Question

Consider the parabolic equation: $$u_t-k(\Delta u+\sum\limits_{i=1}^n a_i\frac{\partial u}{\partial x_i}+bu)=0$$ where $a_i,b,k$ are constants and $k>0$. How this equation can be transformed to the heat equation $w_t-\Delta w=0$.

I tried to use substitution to convert it to heat equation. But I found the solution to the problem $$u_t-k(\sum\limits_{i=1}^n a_i\frac{\partial u}{\partial x_i}+bu)=0$$ is hard to represent.

Is there any easy ways to do this? Thanks so much!

Answer 1

The change $u=e^{-kbt}v$ will get rid of the $0$ order term. Then the change $w=v(x-k\,a_1,x_1,\dots,x-k\,a_n\,x_n,t)$ will get rid of the first order terms.