I am given the following PDE to solve where $b \in \mathbb{R}^{n}$ and $a \in \mathbb{R}$
$u_t+b \cdot Du+cu=0$ in $\mathbb{R}^n\times(0,\infty)$
$u= g$ on $\mathbb{R}^n\times\{t=0\}$
I am told to either solve the PDE using the method of characteristics or by converting the equations into the transport equation by multiplying by a suitable exponential.
I am a bit confused how to apply the method of characteristics. I started followin the method outlined here: http://web.stanford.edu/class/math220a/handouts/firstorder.pdf starting on page 24.
In my problem $F(\vec{x},u,Du)=u_t+b \cdot Du+cu=0$ where
$u=g$ on $\Gamma$.
After parametrizing gamma by a vector $\vec{r}$ so that $\Gamma(\vec{r})=(\gamma_i(\vec{r}),...,\gamma_n(\vec{r}))$, we rewrite our cauchy problem as $F(\vec{x},z,\vec{p})=0$. Then we have exactly 2n+1 characteristic equations given by $$\frac{d x_i}{ds}=F_{p_i}=b_i\\ \frac{dz}{ds}=\sum_{i=1}^{n}p_iF_{p_i}=\sum_{i=1}^{n}b_i\frac{\partial u}{\partial x_i}\\ \frac{dp_i}{ds}=-F_{x_i}-F_zp_i=-cp_i $$ for $i=1,...,n$. Our inital conditions are given by $$x_i(\vec{r},0)=\gamma_i(\vec{r})\\ x(\vec{r},0)=g(\vec{r})\\ p_i(\vec{r},0)=\Psi_i(\vec{r})$$ This is where I am having difficulty. I know I need to solve for $\gamma_i$, $\Psi_i$ by satisfying the following equations $$g_{r_i}=\Psi_1(\vec{r})\frac{\partial\gamma_1}{\partial r_i}+...+\Psi_n(\vec{r})\frac{\partial\gamma_n}{\partial r_i}, i=1,...,n-1\\F(\gamma_i(\vec{r}),..,\gamma_n(\vec{r}),g(\vec{r}),\Psi_1(\vec{r}),..,\Psi_n(\vec{r}))=0.$$ but I'm not sure how. Then I need to solve the system of characteristic equations using the boundary conditions to determine $z$, and then assuming the inital conditions are noncharacteristic the PDE is solved by $u(\vec{x})=z(\vec{r},s)$. I am just confused on how to determine the initial conditions and solve the system of equations corresponding to this problem.