On a nonlocal diffusion problem with mixed Neumann–Dirichlet boundary conditions

We study a nonlocal diffusion equation with a nonlinear Neumann-type inflow on one portion of the boundary and a nonlocal Dirichlet prescription implemented through an exterior interaction region adjacent to the complementary boundary portion. This exterior-volume formulation makes the Dirichlet datum enter the bulk evolution directly and removes the trace-compatibility ambiguity that arises when values are prescribed only on a set of zero N -dimensional measure. Under boundedness, positivity, symmetry, and local Lipschitz assumptions, we prove local existence and uniqueness by Banach's fixed-point theorem, establish a continuation criterion and global solvability under linear growth, and derive continuous dependence, positivity, a comparison principle, a variation-of-constants formula, and an exact mass balance. A one-dimensional quadrature semidiscretization is then analyzed and illustrated by a reproducible computation with residual diagnostics.