Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold

Abstract

A class of integro-differential aggregation equations with nonlinear parabolic term is considered on a compact Riemannian manifold . The divergence term in the equations can degenerate with loss of coercivity and may contain nonlinearities of variable order. The impermeability boundary condition on the boundary of the cylinder is satisfied if there are no external sources of ‘mass’ conservation, . In a cylinder for a sufficiently small , the mixed problem for the aggregation equation is shown to have a bounded solution. The existence of a bounded solution of the problem in the cylinder is proved under additional conditions. For equations of the form with the Laplace-Beltrami operator and an integral operator , the mixed problem is shown to have a unique bounded solution. Bibliography: 26 titles.