No abstract

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We consider the Cauchy problem (1)where N 2 1, p > 1 and u,(x) is a continuous, nonnegative and bounded function. It is well known that the solution u(x,t) of (I), (2) may blow up in a finite time T < +w. We shall be concerned here with the asymptotic behaviour of u(x,t) as blow up is approached.In particular the final profile of u(x, T) near blow up points is studied, and the fact that the blow up set has zero Lebesgue measure is proved.

In this paper, we study mathematical properties of an integro-differential equation that arises as a particular limit case in the study of individual cell-based model. We obtain global wellposedness for some classes of interaction potentials and finite time blow-up for others. The existence of space homogeneous steady states as well as long-time asymptotics for the solutions of the problem is also discussed.

This work is concerned with the system (equation: see text), where Gamma, chi are positive constants and Omega is a bounded and smooth open set in IR2. On the boundary delta Omega, we impose no-flux conditions: (equation: see text). Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x,t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t = 0 for u(x,t), v(x,t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r,t) --> A delta (y) as t --> T for some T < infinity, where A is the total concentration of the species.