## 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)

### MSC:

 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

### Keywords:

sensitivity functions

Zbl 0902.35010