On the uniqueness of a weak solution to 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 ut -∆A(x,u) + div(u∇K *u) = 0 describing the evolution of a colony of bacteria in a bounded convex domain Ω. In this paper we prove the existence and uniqueness of the solution to a mixed problem for a more general equation β(x,u)t = div(∇A(x,u)-β(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∇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∫︀Ω u(x,t)dx = const of bacteria and employed the convexity of Ω 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 Ω.