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.enter image description here

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

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

The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.


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