On unique solvability of nonlocal drift–diffusion-type problems Original Research Article

We prove a priori estimates in L2(0,T;W1,2(Ω)) and L∞(QT), existence and uniqueness of solutions to Cauchy–Neumann problems for parabolic equations equation(0.1) View the MathML source View the MathML source, where ρ(u)=∂σ(u)/∂u>0 and the function v is defined by the nonlocal expression equation(0.2) View the MathML source instead of solving an elliptic boundary problem as in the corresponding local case. Such problems arise as mathematical models of various diffusion–drift processes driven by gradients of local particle concentrations and nonlocal interaction potentials. An example is the transport of electrons in semiconductors, where u has to be interpreted as chemical and v as electro-statical potential.