Resumen:
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The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=?u???(u?v), 0=?v+(u?1), on ??R2, where ? is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)?0, v(x,0)=v0(x)?0 for x??. Under these conditions, the authors prove there exists a radially symmetric solution u(r,t) which blows up at r=0, t=T. A specific description of such a solution is presented. The authors also discuss the strong similarity between the chemotactic model they study and the classical Stefan problem.
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