Characteristics crossing

Consider the Cauchy problem $u\cdot u_x+u_y=1$ for $(x,y)\in \mathbb{R}\times (0,\infty )$, with u(x, 0) = kx, $x\in \mathbb{R}$, for a real parameter k. For which of the following values of k does the problem have a solution defined on $\mathbb{R}\times (0,\infty )$?