Dynamic And Steady States For Multi-dimensional Keller-segel Model With The Diffusion Exponent M>0

Chemotaxis refers to several mechanisms through which cells can move in response to anexternal(usually chemical) signal. Several biological devices allow these small organisms tomove, such as Amebae which is famous because they can aggregate under food restrictions, theprokaryotic has the flagella which is activated by a’bimotor’ which responds to some chemicalsignal. How to observe collective patterns from individual responses to a mean signal emitted bythe cell themselves has attracted very much attention among mathematicians and biologists. Infact, these models share the same mathematical structure, namely the Keller-Segel model whichhas been developed by Keller-Segel in1970s.The competition on signs allows for a competition between the dissipative term and thedrift term which leads to the global existence and finite time blow-up. Although, this system hasbeen widely studied on the global existence and blow-up since2000, there are still no generalstandards on the initial restrictions to determine the global existence and finite time blow-up.Therefore, this thesis will study the global existence and finite time blow-up with diferent initialdata.The main results include the global existence and finite time blow-up for the supercriticalcase, the global existence for the subcritical case, some important properties for the steady states.To be precise:For the supercritical case0<m<22/d, we give a universal constant such that ifthe initial Ld(2m)/2norm is less than the universal constant, then the solution will existglobally.When m>12/d and Ud m)/20∈L(2(Rd), based on the global existence, this thesis showsthat the weak solution satisfies the hyper-contractivity in L∞(Rd).It is shown that the energy critical exponent m=2d/(d+2) is the critical exponent sepa-rating finite Ld(2m)/2norm and infinite Ld(2m)/2norm of the steady state solutions. Morespecifically, when1<m<2d/(d+2), the steady solution satisfies Us d(2m)/2=∞; Ifm>2d/(d+2), then Us d(2m)/2is a constant only depending on the dimension d.A conjecture that the Ld(2m)/2norm for the steady solutions sharply separates the global existence and finite time blow-up is given, that’s if the initial data satisfies U0d(2m)/2<Us d(2m)/2, then the solution exists globally; if the initial data satisfies U0d(2m)/2>Us d(2m)/2, then the solution will blow up at finite time. Our numerical simulation verifiesthis conjecture.