In this paper we give a proof of the existence and uniqueness of smooth solutions for the nonlinear semiconductor Boltzmann equation. The method used allows to obtain global existence in time and uniqueness for dimensions 1 and 2. For dimension 3 we only can assure local existence in time and uniqueness. First, we define a sequence of solutions for a linearized equation and then, we prove the strong convergence of the sequence in a suitable space. The method relies in the use of interpolation estimates in order to control the decay of the solution when the wave vector goes to infinity.
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Jean-Claude Guillot, Global Existence of Solutions for the Nonlinear Boltzmann Equation of Semiconductor Physics. Rev. Mat. Iberoam. 6 (1990), no. 1, pp. 43–59DOI 10.4171/RMI/94