# Uniqueness of solution based on characteristic curves

I have a pde $$\begin{cases} u_t − xu_x = 2u & x\in\mathbb{R}, t>0\\ u(x, 0) = \frac{1}{1+x^2} \end{cases}$$

I've solved it using method of characteristics ($$u=\frac{1}{1+x^2e^{2t}}e^{2t})$$ and plotted charactersitic curves.

Consider the upper half-space since $$t>0$$. How to argue using the drawing whether or not it is the unique solution? Thank you.

• Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$? – Mattos Nov 19 '18 at 1:46
• @Mattos This is the projection on the $(x,t)$-plane – dxdydz Nov 19 '18 at 2:01