On the global existence of solutions to an aggregation model

Abstract In this paper we consider a reaction–diffusion–chemotaxis aggregation model of Keller–Segel type with a nonlinear, degenerate diffusion. Assuming that the diffusion function f ( n ) takes values sufficiently large, i.e. takes values greater than the values of a power function with sufficiently high power ( f ( n ) ⩾ δ n p for all n > 0 , where δ > 0 is a constant), we prove global-in-time existence of weak solutions. Since one of the main features of Keller–Segel type models is the possibility of blow-up of solutions in finite time, we will derive the uniform-in-time boundedness, which prevents the explosion of solutions. The uniqueness of solutions is proved provided that some higher regularity condition on solutions is known a priori. Finally, computational simulation results showing the effect of three different types of diffusion function are presented.

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