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

A class of integro-differential aggregation equations with nonlinear parabolic term $b(x,u)_t$ is considered on a compact Riemannian manifold $\mathscr M$. 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 $\partial\mathscr M\times[0,T]$ of the cylinder $Q^T=\mathscr M\times[0,T]$ is satisfied if there are no external sources of ‘mass’ conservation, $\int_\mathscr Mb(x,u(x,t)) d\nu=\mathrm{const}$. In a cylinder $Q^T$ for a sufficiently small $T$, 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 $Q^\infty=\mathscr M\times[0,\infty)$ is proved under additional conditions. For equations of the form $b(x,u)_t=\Delta A(x,u)-\operatorname{div}(b(x,u)\mathscr G(u))+f(x,u)$ with the Laplace-Beltrami operator $\Delta$ and an integral operator $\mathscr G(u)$, the mixed problem is shown to have a unique bounded solution. Bibliography: 26 titles.

aggregation equation on a manifold, existence of a solution, uniqueness of a solution