Abstract
This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data.
Highlights
In this work, we analyze qualitative properties of non-negative solutions for the chemotaxis system in dimension n ≥ 3 with linear diffusion given by ut = ∆u − ∇ · + uα 1 − Rn uβdx, u(x, 0) = u0(x) ≥ 0, x ∈ Rn, t > 0, x ∈ Rn. (1)Here v(x, t) expresses the chemical substance concentration and it is given by the fundamental solution u(y, t) v(x, t) = K ∗ u(x, t) = cn Rn |x − y|n−2 dy, (2) where πn/2 cn =, n(n − 2)bn bn, Γ(n/2 + 1) (3)bn is the volume of n-dimensional unit ball
V(x, t) expresses the chemical substance concentration and it is given by the fundamental solution u(y, t) v(x, t) = K ∗ u(x, t) = cn Rn |x − y|n−2 dy, (2)
Bn is the volume of n-dimensional unit ball
Summary
N+2 n+2 for any initial data satisfying (4), problem (1) possesses a unique global strong solution defined by Definition 1.1 which is uniformly bounded, i.e., for any t > 0, u(·, t) Lq(Rn) ≤ C1( u0 Lq(Rn)), q ∈ [β + α − 1, ∞], (6) As to (10) and (8), the most remarkable difference is that the mass conservation holds for (10) but not for (8), using this property it’s been proved that the solution of (10) exists globally with small initial data [2, 4, 12, 19, 20], while We firstly state some lemmas which will be used in the proof of local existence and Theorem 2.
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