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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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