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This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant
{ut=d1Δu???(uχ1(w)?w)+μ1u(1?u?a1v),xΩ,t>0,vt=d2Δv???(vχ2(w)?w)+μ2v(1?a2u?v),xΩ,t>0,wt=d3Δw?(αu+βv)w,xΩ,t>0
under homogeneous Neumann boundary conditions in a bounded domain Ω?Rn (n1) with smooth boundary, where the initial data (u0,v0)(C0(Ω))2 and w0W1,(Ω) are non-negative and the parameters d1,d2,d3>0, μ1,μ2>0, a1,a2>0 and α,β>0. The chemotactic function χi(w) (i=1,2) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for i=1,2,(i) χi(w)=χ0,i>0 and
6w06L(Ω)<πdid3n+1χ0,i?2did3n+1χ0,iarctan?di?d32n+1did3;
(ii) 0<6w06L(Ω)d33(n+1)6χi6L[0,6w06L(Ω)]min?{2didi+d3,1}.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If a1,a2(0,1) and u00v0, then any global bounded solution exponentially converge to (1?a11?a1a2,1?a21?a1a2,0) as t;? If a1>1>a2>0 and v00, then any global bounded solution exponentially converge to (0,1,0) as t;? If a1=1>a2>0 and v00, then any global bounded solution algebraically converge to (0,1,0) as t.  相似文献   

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In this short Note we give a self-contained example of a consistent family of holomorphic semigroups (Tp(t))t?0 such that (Tp(t))t?0 does not have maximal regularity for p>2. This answers negatively the open question whether maximal regularity extrapolates from L2 to the Lp-scale.  相似文献   

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Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let Sn be the symmetric group on {1,2,,n}, and let S={si|1in?1} be the generating set of Sn, where for 1in?1, si is the adjacent transposition. For a subset J?S, let (Sn)J be the parabolic subgroup generated by J, and let (Sn)J be the set of minimal coset representatives for Sn/(Sn)J. For uv(Sn)J in the Bruhat order and x{q,?1}, let Ru,vJ,x(q) denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for Ru,vJ,x(q) when J=S?{si}, and obtained an expression for Ru,vJ,x(q) when J=S?{si?1,si}. In this paper, we provide a formula for Ru,vJ,x(q), where J=S?{si?2,si?1,si} and i appears after i?1 in v. It should be noted that the condition that i appears after i?1 in v is equivalent to that v is a permutation in (Sn)S?{si?2,si}. We also pose a conjecture for Ru,vJ,x(q), where J=S?{sk,sk+1,,si} with 1kin?1 and v is a permutation in (Sn)S?{sk,si}.  相似文献   

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《Discrete Mathematics》2006,306(19-20):2438-2449
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