Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species. (English) Zbl 1265.35135

Espejo, E. E.; Stevens, A.; Suzuki, T.

Differ. Integral Equ. 25, No. 3-4, 251-288 (2012).

Summary: The authors consider the interacting system of Smoluchowski-Poisson equations \[ \partial _t u_1 = d_1 \Delta u_1 - \chi _1 \nabla \cdot u_1 \nabla v,\;\partial _t u_2 = d_2 \Delta u_2 - \chi _2 \nabla \cdot u_2 \nabla v \;\text{ in }\;\Omega \times (0,T) \]
\[ d_1 \frac {\partial u_1}{\partial \nu } - \chi _1 u_1 \frac {\partial v}{\partial \nu }= d_2 \frac {\partial u_2}{\partial \nu } - \chi _2 u_2 \frac {\partial v}{\partial \nu } \;\text{ on }\;\partial \Omega \times (0,T) \] with the initial condition \(u_{01} \geq 0,\;u_{02} \geq 0\) in \(\Omega \) and \[ -\Delta v = u_1+u_2 - \frac {1}{| \Omega | } \int _\Omega (u_1+u_2) dx,\;\frac {\partial v}{\partial \nu } = 0 \text{ on }\;\partial \Omega ,\;\int _\Omega v dx =0. \] Here \(\Omega \) is a bounded two dimensional domain with a smooth boundary, \(d_1\), \(d_2\), \(\chi _1\), \(\chi _2\) are positive parameters. It is proved that for a certain region of parameters there is a simultaneous blow-up in a finite time \(T\), i.e. \(\lim _{t\to T-} \| u_1(t,.)\| _\infty =\lim _{t\to T-} \| u_2(t,.)\| _\infty = +\infty \), even for non-radially symmetric case. For a finite existence time of the solution, a formation of collapse (possibly degenerate) for each component, total mass quantization and formation of sub-collapses are described. For the case of radially symmetric solutions, the authors prove rigorously that the collapse concentrates mass on one component if the total mass of the other component is relatively small.

MSC:

35K55	Nonlinear parabolic equations
92C17	Cell movement (chemotaxis, etc.)

Keywords:

Smoluchowski-Poisson equations; chemotaxis; simultaneous blow-up; formation of collapse; mass separation