Quasilinear Keller Segel System Research Articles

Global-in-time bounded weak solutions to a degenerate quasilinear Keller–Segel system with rotation

We consider a degenerate quasilinear Keller–Segel system of fully-parabolic type involving rotation in the aggregative term,0 \\end{array}\\right. \\end{equation*} ?>where is a bounded convex domain with smooth boundary. Here S(u, v, x) = (si, j)2×2 is a matrix with . Moreover, for all with nondecreasing on [0, ∞). It is shown that whenever m > 1, for any non-negative initial data, which is sufficiently smooth, the system possesses global and bounded weak solution.

Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains

This paper deals with the quasilinear fully parabolic Keller–Segel system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN with smooth boundary, N∈N. The diffusivity D(u) is assumed to satisfy some further technical conditions such as algebraic growth and D(0)⩾0, which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that S(u)/D(u)⩽K(u+ε)α for u>0 with α<2/N, K>0 and ε⩾0. When D(0)>0, this paper represents an improvement of Tao and Winkler [17], because the domain does not necessarily need to be convex in this paper. In the case Ω=RN and D(0)⩾0, uniform-in-time boundedness is an open problem left in a previous paper [7]. This paper also gives an answer to it in bounded domains.

A lower bound for blow-up time in a fully parabolic Keller–Segel system

In this paper, an explicit lower bound for the blow-up time is obtained to a parabolic–parabolic Keller–Segel system, the blow-up conditions of which were established with an upper bound of blow-up time by Cieślak and Stinner [T. Cieślak, C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations 252 (2012) 5832–5851].

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.

Extinction, decay and blow-up for Keller–Segel systems of fast diffusion type

We consider the quasi-linear Keller–Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm LN(q−m)2 for all dimensions N⩾2 and all exponents q⩾2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<∞ for 1−2N⩽m<1 with the mass conservation is shown. On the other hand, in the case of 0<m<1−2N, the extinction phenomenon of solution is proved. It is clarified that the case of m=1−2N exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1−2N. For the borderline case of m=1−2N, our solution decays in Lr exponentially as t→∞.