I came across the following PDE: $$ \partial_xf(x,y)+\partial_y(f(x,y))=\gamma(x)(f^2(x,y)-f(0,y)) $$ Assume that $f$ is a well behaved function and $\gamma(x)$ is given and I have the initial conditions $f(0,y)=g(y)$.
I have worked several times with the characteristic curves method and I know the separation-variables methods, I attacked it with Laplace and Fourier transforms but nothing got out of them.