On uniqueness of weak solution to mixed problem for integro-differential aggregation equation

Abstract

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 $$ u_t - \Delta A(x, u) + {\rm div}\, (u\nabla K \ast u)=0 $$ 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 $$ \beta(x,u)_t={\rm div}\,(\nabla A(x,u)-\beta(x,u)G(u))+f(x,u). $$ 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 $\nabla K \ast u$. 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 $\int_\Omega u(x,t)dx=const$ 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 $\Omega$.