Solution to the cauchy problem: $u^{2}_{x}/2−u_{y} = −x^{2}/2 , u(x, 0) = x$

Question

$\frac 12 u^{2}_{x}−u_{y} = −\frac 12x^{2} ;\\ u(x, 0) = x$

How do we show that that the solution blows up in finite time and explain this in terms of characteristic of the equation.