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.