In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.

We consider a chemotaxis system over a bounded domain $ \Omega\subset \mathbb R^2 $ of the following form $\begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)- u+ a, \\ &v_t = \Delta v- v+ uv(1-v)+b, \end{split} \right. \ \ \ \ \ \ (\star)\end{equation*}$ with $ \chi>0 $, $ a\geq0 $ and $ b>0 $. The particular version of ($ \star$) with $ \chi = 2 $ was proposed by Pitcher to describe the evolution of population dynamics of criminal agents. Recent results reveal that the system ($ \star$) associated with Neumann boundary conditions admits a global classical solution, under appropriate smallness conditions on both the initial data and the parameter $ \chi $. The present study indicates that nevertheless, for all reasonably regular initial data and any $ \chi>0 $, the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Furthermore, it also demonstrates that, whenever $ a = 0 $, such global generalized solution becomes bounded and smooth at least eventually. In particular, it approaches the spatial equilibria at exponential rate in the large time limit.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.