Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with a logistic source

“…The following local existence result is rather standard, since a similar reasoning in [3,7,24,28,29,30,38]. We omit it here.…”

Global boundedness in a quasilinear chemotaxis model with nonlinear diffusion and consumption of chemoattractant

“…First, we state the local‐in‐time existence result of a classical solution of , which is similar to the ones in previous studies. ()…”

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

“…For m = α =1 and χ ( v )= χ >0, it is proved in Tello and Winkler that when μ >( n −2) χ / n the solutions are globally bounded, whilst for , α =1 and χ ( v )= χ >0 the same result is achieved in Cao and Zheng under the assumption μ >(1−2/( n (2− m ) + )) χ . In Zheng, the author formulates problem under the assumptions m ≥1, α >0 and χ ( v )= χ >0, and with a more general expression for the logistic absorption: k u − μ u r , with r >1. It is concluded that the coefficient μ does not take part for the boundedness of the solution when , whilst it does for α +1= r .…”

“…Specifically, parameter sweeps of m , α , and n are used to detect critical exponents which delineate the regions in which u is, and is not, bounded. Critically, although the inequalities on α , herein derived, and from Zheng, ensure the boundedness of u , regardless the size of the dampening term, μ , in the logistic source, these inequalities are not tight. Explicitly, we are able to violate them and still produce bounded simulations.…”