ON UNIQUENESS OF WEAK SOLUTION TO MIXED PROBLEM FOR INTEGRO-DIFFERENTIAL AGGREGATION EQUATION

In a well-known paper by A. Bertozzi, D. Slepcev (2010), there was established the existence and uniqueness of solution to a mixed problem for the aggregation equation u(t) - Delta A(x, u) + div (u del (*) u) = 0 describing the evolution of a colony of bacteria in a bounded convex domain Omega. In this paper we prove the existence and uniqueness of the solution to a mixed problem for a more general equation beta(x, u)(t) = div(del A(x, u) - beta(x, u)G(u)) + f(x, u). The term f(x, u) in the equation models the processes of "birth-destruction" of bacteria. The class of integral operators G(v) is wide enough and contains, in particular, the convolution operators del K (*) u. The vector kernel g(x, y) of the operator G(u) can have singularities. Proof of the uniqueness of the solution in the work by A. Bertozzi, D. Slepcev was based on the conservation of the mass integral(Omega) u(x, t)dx = const of bacteria and employed the convexity of Omega and the properties of the convolution operator. The presence of the "inhomogeneity" f (x, u) violates the mass conservation. The proof of uniqueness proposed in the paper is suitable for a nonuniform equation and does not use the convexity of Omega.