= Solution
For real initial data, <separation of variables> gives
$$
\boxed{z(t)=\frac{z_0}{1-z_0t}.}
$$
This includes $z_0=0$, whose solution is identically zero. Every other real $z_0$ has a pole at $t=1/z_0$, so only $z_0=0$ gives a finite solution on the whole real line. If complex-valued initial data are allowed, a nonreal $z_0$ has no real-time pole; the whole-real-line conclusion here uses the usual real-valued differential-equation convention.
Along a <characteristic curve> $y=u+ax$, the <chain rule> gives $d f(x,u+ax)/dx=f_x+af_y$. For the homogeneous <transport equation> this is zero, so $f(x,y)=F(y-ax)$. For the nonlinear equation, the same <method of characteristics> reduces the problem to $d f/dx=f^2$, with initial value $g(u)$. Therefore
$$
\boxed{f(x,y)=\frac{g(y-ax)}{1-xg(y-ax)}.}
$$
For <differentiable> real $g$, this is the classical solution wherever its denominator is nonzero, on the characteristic intervals containing the initial line. If $g(u)\ne0$ for some $u$, that characteristic blows up at $x=1/g(u)$, $y=u+a/g(u)$. Thus \b[the only real-valued solution bounded on all of $\mathbb R^2$ has $g\equiv0$, giving $f\equiv0$.]
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