Scalar conservation law in a bounded domain with strong source at boundary

We consider a scalar conservation law with source in a bounded open interval $\Omega \subseteq \mathbb R$. The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function $\varrho$ with an intensity function $V: \Omega \rightarrow \mathbb R_+$ that grows to infinity at $\partial \Omega$ . We define the entropy solution $u \in L^\infty$ and prove the uniqueness. When V is integrable, u satisfies the boundary conditions introduced by F. Otto (C. R. Acad. Sci. Paris, 322(1):729–734, 1996), which allows the solution to attain values at $\partial \Omega$ different from the given boundary data. When the integral of V blows up, u satisfies an energy estimate and presents essential continuity at $\partial\Omega$ in a weak sense.