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Finite-time aggregation into a single point in a reaction-diffusion system. (English) Zbl 0909.35071
Author’s abstract: We consider the following system: \[ u_t= \Delta u-\chi\nabla(u\nabla v),\quad \chi>0,\quad \Delta v= 1-u,\tag{S} \] which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension \(N=3\), (S) possesses radial solutions that blow-up in a finite time. The asymptotic behavior of such solutions is analyzed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behavior and that known to occur for blowing-up solutions of (S) in the case \(N= 2\) are also discussed.

MSC:
35K57 Reaction-diffusion equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K45 Initial value problems for second-order parabolic systems
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