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Concentration lemma, Brezis-Merle type inequality, and a parabolic system of chemotaxis. (English) Zbl 1009.35043

Summary: We study the system \[ \begin{aligned} u_t= \nabla\cdot(\nabla u-\chi u\nabla v)\quad &\text{in }\Omega\times (0,T),\\ \tau v_t=\Delta v-\gamma v+\alpha u\quad &\text{in }\Omega\times (0,T),\\ \partial u/\partial n=\partial v/\partial n= 0\quad &\text{on }\partial\Omega\times (0,T),\\ u|_{t=0}= u_0,\;v|_{t=0}= v_0\quad &\text{in }\Omega,\end{aligned} \] introduced by E. F. Keller and L. A. Segel to describe the chemotactic feature of slime molds. Concentration towards the boundary is shown for the blow-up solution with the total mass less than \(8\pi\). For this purpose, a variant of the concentration lemma of Chang and Yang’s type, and also a parabolic version of an inequality due to Brezis and Merle, are provided.

MSC:

35K57 Reaction-diffusion equations
92C17 Cell movement (chemotaxis, etc.)
35Q80 Applications of PDE in areas other than physics (MSC2000)
92C15 Developmental biology, pattern formation
35K45 Initial value problems for second-order parabolic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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