A boundary value problem for a quasilinear equation of first order with arbitrary dependence of the direction of the characteristics at the boundary on the unknown function

A boundary value problem for the equation
$$ \frac d{dx_k}a_k(x,u)+b(x,u)+cu=0 $$
is posed and investigated in a domain $\Omega\subset\mathbf R^n$ with boundary $S$. Let $a_\nu$ be the normal component on $S$ of the vector $\vec a=(a_1,\dots,a_n)$. In contrast to previous papers, an arbitrary dependence of $a_\nu(x,u)$ on $u$ is permitted.
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