On a nonlocal aggregation model with nonlinear diffusion
This paper concerns a nonlocal aggregation equation with nonlinear degenerate diffusion which arises from the study of biological aggregation dynamics: \[ \partial_t u + \partial_x(uv) = 0\;\text{ for } (t,x) \in (0,\infty) \times \mathbb R, \quad\text{and}\quad u(0,x) = u_0(x)\;\text{ for } x \in \mathbb R, \] where \(v = \partial_x(K * u) - ru\partial_xu\), \(K= e^{-|x|}\), \(r\) is a positive constant, and \(u_0 \in \bigcap_{m=0}^\infty H^m(\mathbb R)\). The solution \(u = u(t,x)\) represents population density in biology or particle density in material science. This equation is derived by \textit{Bertozzi, Lewis} and \textit{Topaz} [Bull. Math. Bio. 68, 1601--1623, (2006)]. By using the \(L^\infty\) energy method introduced by \textit{M. Ôtani} and \textit{Y. Sugiyama} [J. Math. Soc. Japan 53, No.~4, 745--789 (2001; Zbl 1024.35056)], the authors show the existence and uniqueness of smooth local solutions for initial data which is not necessarily nonnegative. It is also proved that any nonzero smooth initial data with compact support will lead to blowup of the gradient in finite time.
scientific article