The phenomenon of large population densities in a chemotaxis competition system with loop
- PDF / 584,050 Bytes
- 38 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 173 Views
Journal of Evolution Equations
The phenomenon of large population densities in a chemotaxis competition system with loop Xinyu Tu, Chun- Lei Tang and Shuyan Qiu
Abstract. We study herein the initial boundary value problem for a two-species chemotaxis system with loop ⎧ ∂t u 1 = u 1 − χ11 ∇ · (u 1 ∇v1 ) − χ12 ∇ · (u 1 ∇v2 ) + μ1 u 1 (1 − u 1 − a1 u 2 ), x ∈ , ⎪ ⎪ ⎨ ∂t u 2 = u 2 − χ21 ∇ · (u 2 ∇v1 ) − χ22 ∇ · (u 2 ∇v2 ) + μ2 u 2 (1 − u 2 − a2 u 1 ), x ∈ , ∂t v1 = v1 − v1 + u 1 + u 2 , x ∈ , ⎪ ⎪ ⎩ ∂t v2 = v2 − v2 + u 1 + u 2 , x ∈ ,
competition
t t t t
> 0, > 0, > 0, > 0,
under the homogeneous Neumann boundary condition, where ⊂ Rn (n ≥ 3) is a smooth and bounded domain, χi j > 0, μi > 0, ai > 0 (i, j = 1, 2). In the radial symmetric setting, for any T > 0 and L > 0, it is proved that there exists positive initial data such that the corresponding solution (u 1 , u 2 , v1 , v2 ) satisfies u 1 (x L , t L ) > L or u 2 (x L , t L ) > L for some (x L , t L ) ∈ × (0, T ). Moreover, when χ11 = χ12 , χ21 = χ22 , μ = max{μ1 , μ2 } ∈ (0, 1), one can find initial 2 2 data (u 10 , u 20 , v10 , v20 ) ∈ C 0 () × W 1,∞ () , which is irrelevant to μ, such that for all μ ∈ (0, 1), the corresponding solution (u 1,μ , u 2,μ , v1,μ , v2,μ ) fulfills u 1,μ (xμ , tμ ) >
L L or u 2,μ (xμ , tμ ) > for some (xμ , tμ ) ∈ × (0, T ). μ μ
In particular, it is proved that blowup for one of the chemotactic species implies also blowup for the other one at the same time.
1. Introduction Chemotaxis, the property such that species move toward or away from the higher concentration of the chemical substance, has attracted many mathematicians’ interests in recent years. So far, there is a considerable amount of research on the classical Keller–Segel system [10] ∂t u = u − χ ∇ · (u∇v) + f (u), x ∈ , t > 0, (1.1) x ∈ , t > 0 τ ∂t v = v − λv + αu k , Mathematics Subject Classification: 35B44, 35K55, 92C17 Keywords: Chemotaxis with loop, Two species and two stimuli, Lotka–Volterra-type competition, Simultaneous blowup.
X. Tu et al.
J. Evol. Equ.
with τ ∈ {0, 1}. For the case f (u) = 0, k = 1, there is no possibility that the solution of (1.1) blows up when n = 1; when n = 2, the system of (1.1) exhibits a remarkable feature: critical mass [6,7,23,24]. As for the case n ≥ 3, one can refer to [42,44] for the blowup results. For the case f (u) = r u − μu 2 , k = 1, when n = 2, it was proved that the solution of (1.1) with τ = 1 is globally bounded for any r ∈ R and μ > 0 [27]; in the smooth and bounded convex domain ⊂ Rn (n ≥ 3), Lankeit obtained the existence of global weak solutions to (1.1) with τ = 1 for arbitrary μ > 0 [12], especially, in the three-dimensional setting, they further proved that these solutions become classical solutions after some time; what is more, under the condition that μ is sufficiently large, the large time behavior of solutions to (1.1) with τ = 1 was considered in the convex domain [45]. For the case f (u) ≤ a − bu 2 , k = 1, in the convex domain ⊂ Rn (n ≥ 3), the global solvability for τ
Data Loading...
