UMD PDE Qualifying Exams/EvansCh2

Problem 1

Write an explicit formula for a function $u$ solving the initial-value problem $\{\begin{array}{cc}{u}_t+b\cdot Du+cu=0& \text{ in }{ℝ}^n\times (0,\mathrm{\infty })\\ u=g& \text{ on }{ℝ}^n\times \{t=0\}.\end{array}$ where $c\in ℝ$ and $b\in {ℝ}^n$ are constants.

Solution

Consider characteristics $(x(s),t(s))=(x_0+bs,s)\in {ℝ}^n$. Also, for any $x\in {ℝ}^n,t\in ℝ$, 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 $x_0=x-tb$ and hence

$u(x,t)=g(x-tb)e^{-ct}$.