Q:

Characteristics method applied to the PDE $u_x^2 + u_y^2=u$

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sense (sum of squares is $0$ and all constants are $0$). I would be grateful for a reasonable explanation how to solve that with help of characteristics.

A:

I don't see how the method of characteristics (as I know) could be applied to an equation that involves the derivatives in a nonlinear way. 
Since $u\ge 0$, we can look for $u$ in the form $u=v^2$ which leads to a simplification: by the chain rule, 
$$u_x^2+u_y^2 = 4v^2(v_x^2+v_y^2)$$ 
so in terms of $v$ the PDE becomes 
$$
v_x^2+v_y^2 = \frac14 \tag{1}
$$
This is a form of the Eikonal equation, the solution of which is $1/2$ times the distance function from the set $\{v=0\}$. 
However, the boundary condition is incompatible with the PDE. Indeed, it asserts that $v(x,0)=x$. But according to (1), $|\nabla v|=1/2$ which implies the Lipschitz bound $$|v(a)-v(b)|\le \frac12|a-b|\tag{2}$$ The Lipschitz bound, being a pointwise inequality, must extend to the boundary values; however, $v(x,0)=x$ does not satisfy (2).

