Write an explicit formula for a function $u$ solving the initialvalue problem
$\left\{{\begin{array}{r l}u_{t}+b\cdot Du+cu=0&{\text{ in }}\mathbb {R} ^{n}\times (0,\infty )\\u=g&{\text{ on }}\mathbb {R} ^{n}\times \{t=0\}.\end{array}}\right.$
where $c\in \mathbb {R}$ and $b\in \mathbb {R} ^{n}$ are constants.

Consider characteristics $(x(s),t(s))=(x_{0}+bs,s)\in \mathbb {R} ^{n}$ . Also, for any $x\in \mathbb {R} ^{n},t\in \mathbb {R}$ , consider $z(s)=u(x+sb,t+s)$ . Then taking a derivative gives
${\dot {z}}(s)=Du(x+sb,t+s)\cdot b+u_{t}(x+sb,t+s)=cz(s)$
where the last inequality is a result of the original PDE.
The above ODE can be solved and we get
$z(s)=z(0)e^{cs}$
Finally, any point $(x,t)$ is connected to the characteristic curve $(x_{0},0)$ where $x_{0}=xtb$ and hence
$u(x,t)=g(xtb)e^{ct}$ .