UMD PDE Qualifying Exams/EvansCh2

Problem 1

Write an explicit formula for a function $u$ solving the initial-value 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.

Solution

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 $x_{0}=x-tb$ and hence

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