Global Boundedness In A Quasilinear Keller-Segel System With Logistic Source

In this paper, we consider a quasilinear parabolic-parabolic Keller-Segel system with a logistic type source ut= ? · (?(u)?u) - ? · (?(u)?v)+g(u), vt= ?v - v+u in a smooth bounded domain ? (?) Rn, n? 1, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where ?,? and g are smooth positive functions satisfying c1sp??(s) and c1sq??(s)< c2sq for p ?R, q> 0 and s? s0> 1, g(s)? as -?sk for s> 0, with constants a? 0,?, c1, c2> 0, and the extended logistic exponent k> 1 instead of the ordinary k= 2. We mainly discuss how the logistic exponent k effecting the global boundedness of the above system. It is proved that if q< k - 1, or q= k - 1 with ? properly large that ???0 for some ?0> 0, then there exists a classical solution which is global in time and bounded. This shows the exact way of the logistic exponent k> 1 effecting the behavior of solutions.Chapter 1 introduces the background and current progress of the field, and states the background of the original Keller-Segel system, as well as its further development with the related conclusions. In addition, we state our main contents of the thesis. In Chapter 2, we give some auxiliary lemmas and the a prior estimates involved as preliminaries. In Chapter 3, based on the auxiliary lemmas and the a prior estimates of Chapter 2, we give the proof of the main results of the thesis.