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The solutions of the quasilinear Keller-Segel system with the volume filling effect do not blow up whenever the Lyapunov functional is bounded from below

100%
Cieślak T.
Banach Center Publications
|
2006
|
tom 74
|
nr 1
127-132
EN
In [2] we proved two kinds of mechanisms of preventing the blow up in a quasilinear non-uniformly parabolic Keller-Segel systems. One of them was a priori boundedness from below of the Lyapunov functional. In fact, we were able to present a condition under which the Lyapunov functional is bounded from below and a solution exists globally. In the present paper we prove that whenever the Lyapunov functional is bounded from below the solution exists globally.
2
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Global existence of solutions to a chemotaxis system with volume filling effect

100%
Cieślak T.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 1
117-134
EN
A system of quasilinear parabolic equations modelling chemotaxis and taking into account the volume filling effect is studied under no-flux boundary conditions. The resulting system is non-uniformly parabolic. A Lyapunov functional for the system is constructed. The proof of existence and uniqueness of regular global-in-time solutions is given in cases when either the Lyapunov functional is bounded from below or chemotactic forces are suitably weakened. In the first case solutions are uniformly bounded in time, in the second one it is shown that a uniform bound is not possible.
3
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Global existence and convergence to steady states in a chemorepulsion system

51%
Cieślak T., Laurençot P., Morales-Rodrigo C.
Banach Center Publications
|
2008
|
tom 81
|
nr 1
105-117
EN
In this paper we consider a model of chemorepulsion. We prove global existence and uniqueness of smooth classical solutions in space dimension n = 2. For n = 3,4 we prove the global existence of weak solutions. The convergence to steady states is shown in all cases.
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