Global solutions to some parabolic-elliptic systems of chemotaxis. (English) Zbl 0941.35009

Global time solvability is studied for the parabolic-elliptic system: \(u_t=\nabla \cdot(\nabla u-u\nabla \varphi(v))\), in \(\Omega\times \mathbb{R}^+\) with Neumann boundary conditions both for \(u\) and \(v\). The present author generalizes results from [T. Nagai and T. Senba, Adv. Math. Sci. Appl. 8, No. 1, 145-156 (1998; Zbl 0902.35010)] for radial domains separately in three directions, namely more general “sensitivity” functions \(\varphi\), nonradial domains and with an extra diffusion term for \(v\). The main settings are the following. 1) He treats \(\varphi\) for the radial case with \(N=2\) under the conditions \(0<\varphi'(v)\to 0\) for \(v\to\infty\) and \(v \varphi' (v)\) increasing. 2) For \(\varphi(v)= \chi\log v\) with \(0<\chi <2/N\) \((N\geq 2)\), and \(u(\cdot,0) \in L^q(\Omega)\) with \(q>N/2\) he considers the case of general domains. 3) Assuming \(N=2\), \(\varphi(v)= \chi\log v\) with \(0<\chi<1\) and \(\nabla v(\cdot, 0)\in L^2(\Omega)\), the second equation is replaced by \(\varepsilon v_t-\Delta v+v=u\). In all three cases he proves that the solution for the initial value problem exists globally in time. The main step in the proofs is to show an a priori bound for \(u(\cdot,t)\) which uses some delicate nonlinear analysis. The system has been considered by the same author in earlier papers. The present results are rather sharp; slightly different sensitivity functions have displayed blow-up in finite time.
Reviewer: G.H.Sweers (Delft)


35B40 Asymptotic behavior of solutions to PDEs
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations


Zbl 0902.35010