Ayazoglu, Rabil; Ekincioglu, Ismail

This paper deals with the following competitive two-species chemotaxis system with two chemicals with a source term of logistic type ut=Δu−χ1∇⋅(u∇υ)+η1u−μ1u2, υt=Δυ−υ+ω, ωt=Δω−χ2∇⋅(ω∇z)+η2ω−μ2ω2, 0=Δz−z+u, (x,t)∈Ω×(0,T) under homogeneous Neumann boundary conditions [Formula presented] and initial conditions u(x,0)=u0(x), υ(x,0)=υ0(x), ω(x,0)=ω0(x), x∈Ω, in a bounded domain Ω⊂RN (N≥1), where the parameters χ1, μ1, μ2>0 and η1, η2, χ2∈R. It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds: (i)χ2≤0, (ii)χ2>0 as well as μ2 is large enough, which partially improves the results of [25].

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Ayazoglu, Rabil

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of variable logistic type ut=∇⋅(ϕ(u)∇u)−∇⋅(ψ(u)∇υ)+g(x,u), −Δυ=−υ+u, in (x,t)∈Ω×(0,T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in RN (N≥1) and g∈C1(Ω‾×[0,∞)) function satisfies g(x,s)≤ηsk(x)−μsm(x),s>0 with g(x,0)≥0, x∈Ω and η≥0,μ>0 are constants, k,m are measurable functions with 1<k−:=ess infx∈Ωk(x)≤k(x)≤k+:=esssup k(x)x∈Ω<+∞, 2≤m−:=ess infx∈Ωm(x)≤m(x)≤m+:=esssup m(x)x∈Ω<+∞. Positive functions ψ,ϕ∈C2([0,∞)) satisfy νsq≤ψ≤χsq and ϕ≥σsp with p,q∈R, ν,χ,σ>0 when s≥0. It is proved that if q=m−−1 (critical case) and [Formula presented] then there exists a classical solution which is global in time and bounded. Moreover, if q∈(k+−1,m−−1), we obtain that there is a classical solution of the above system uniformly in time bounded without any restrictions on m− and μ.

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