COLLOQU IUM MATHEMAT I CUM VOL NO

COLLOQU IUM MATHEMAT I CUM VOL. 106 2006 NO. 2 GLOBAL EXISTENCE VERSUS BLOW UP FOR SOME MODELS OF INTERACTING PARTICLES BY PIOTR BILER (Wro ªaw) and LORENZO BRANDOLESE (Lyon) Abstra t. We study the global existen e and spa e-time asymptoti s of solutions for a lass of nonlo al paraboli semilinear equations. Our models in lude the Nernst Plan k and DebyeHü kel drift-diusion systems as well as paraboli -ellipti systems of hemotaxis. In the ase of a model of self-gravitating parti les, we also give a result on the nite time blow up of solutions with lo alized and os illating omplex-valued initial data, using a method due to S. Montgomery-Smith. 1. Introdu tion. In this paper we are on erned with semilinear paraboli systems of the form (1) ∂tuj = ∆uj + ? · ( m∑ h,k=1 cj,h,k uh(?Ed ? uk) ) , j = 1, . . . ,m, u(0)(x) = u0(x). Here the unknown is the ve tor eld u = (u1, . . . , um), dened on the whole spa e Rd (with m ≥ 1 and d ≥ 2), and cj,h,k ? L∞(Rd), j, h, k = 1, .

• self gravitating


• i1 ?


• unique solution


• global existence versus blow up


• paraboli systems


• valued


• similar de ay proles