Is the solution of a quasilinear pde unique if you solve it by transforming it into an ode?

Question

$xu_x + yu_y = 4u$ on the unit disk in $\mathbb{R}^2$. With boundary condition $u=1$ on the boundary. And got $u = (x^2+y^2)^2$ by solving it by change of variables. How do I show it is unique? And is this always the case?

Answer 1

To show it is unique, let $u$ and $v$ be solutions of this equation and define $w = u-v$. Then $w$ solves $xw_x+yw_y=4w$ with $w=0$ on the boundary. If you can show that $w$ is identically zero, you are done. This case is pretty easy and the solution is unique.

Nonuniqueness can occur in a couple of ways for equations like these. The easiest to spot is when the coefficients that form the ODE system are not Lipschitz and lead to an ODE system without unique solutions. Another more subtle way that solutions can lack uniqueness is if the boundary consists of a characteristic curve. In this case, the PDE actually doesn't define the solution anywhere outside of the boundary and the solution is not unique outside of this boundary curve.