Large time behavior of nonlocal aggregation models with nonlinear diffusion (Q998850)

Let \(G\) be a smooth and even kernel and \(V\) be a smooth and even external potential, and consider the nonlocal parabolic equation \[ \partial_t \varrho = \partial_x \left( \varrho \partial_x \left( a(\varrho) - G*\varrho + V \right) \right) \;\;\;\text{ in }\;\;\; (0,\infty)\times {\mathbb R}\;, \] the function \(a\) being either zero or a nonnegative increasing function. Well-posedness and finite speed of propagation are first studied. In the absence of diffusion (\(a=0\)), the existence of (measure-valued) steady states is investigated according to the properties of \(G\) and \(V\). Their stability is also analysed in some cases. When \(a(\varrho)=\varepsilon \varrho^2\) and \(\varepsilon\) is small enough, a perturbation argument is used to show the existence of steady states. The analysis relies on an alternative formulation of the problem involving the derivative of the pseudo-inverse of an indefinite integral of \(\varrho\).