Uniqueness of the Weak Solution to a Cross-Diffusion System Without Volume Filling

We consider a system of parabolic partial differential equations with a cross-diffusion phenomenon. Previous results showed that a weak solution exists to the semiconductor model with electron-hole scattering. In this work, we show that this weak solution exists uniquely. For weak solutions of cross-diffusion systems, few uniqueness results have been derived. Among these uniqueness results, we require that weak solutions are bounded. The weak solution of the semiconductor model may not be bounded, so its uniqueness is very difficult to prove. We rely on the structural character of this model to derive a sequence of weak solutions. By considering the limit of this sequence of solutions, we show that the weak solution of the semiconductor model is unique.