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 $\mathrm{\Omega }\subseteq \mathbb{R}$. 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 $V:\mathrm{\Omega }\to {\mathbb{R}}_{+}$ that grows to infinity at $\mathrm{\partial }\mathrm{\Omega }$. We define the entropy solution $u\in L^{\mathrm{\infty }}$ and prove the uniqueness. When $V$ is integrable, $u$ satisfies the boundary conditions introduced in [F. Otto, C. R. Acad. Sci. Paris 1996], which allows the solution to attain values at $\mathrm{\partial }\mathrm{\Omega }$ different from the given boundary data. When the integral of $V$ blows up, $u$ satisfies an energy estimate and presents essential continuity at $\mathrm{\partial }\mathrm{\Omega }$ in a weak sense.