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Blow-up of radially symmetric solutions to a chemotaxis system. (English) Zbl 0843.92007
Cellular slime molds exhibit the oriented movement in response to a certain chemical released by themselves in their environment, and form aggregation. Such oriented movement is called chemotaxis. E. F. Keller and L. A. Segel [J. Theor. Biol. 9, 399-415 (1970)] proposed a mathematical model to describe the initiation of chemotactic aggregation of cellular slime molds. With the cell density $$u(x, t)$$ and the concentration of the chemical $$v(x, t)$$ at place $$x$$ and time $$t$$, a simplified Keller-Segel model considered in this paper is described by the system of the two partial differential equations ${{\partial u} \over {\partial t}}= \nabla\cdot (\nabla u-\chi u\nabla v), \quad \varepsilon {{\partial v} \over {\partial t}}= \Delta v- \gamma v+ \alpha u, \qquad x\in \Omega,\;t>0,$ subject to homogeneous Neumann boundary conditions, where $$\chi$$, $$\varepsilon$$, $$\gamma$$ and $$\alpha$$ are positive numbers and $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ with smooth boundary $$\partial \Omega$$.
We study the global existence and blow-up of solutions in radially symmetric situations. It is shown that blow-up never occurs in one dimension. In two dimensions, we give a threshold number explicitly such that if a total cell number $$\int_\Omega u_0 dx$$ is larger than the threshold number then blow-up can occur. In three or more dimensions, blow-up can occur even though $$\int_\Omega u_0 dx$$ is small.

##### MSC:
 92C40 Biochemistry, molecular biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 92C99 Physiological, cellular and medical topics