GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE

This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0 vt=Δv+w−v,x∈Ω,t>0,wt=u−w,x∈Ω,t>0,in a bounded domainΩ⊂R^n(n≥2)with smooth boundary ∂Ω,where the diffusion coefficient D(u)and the chemotactic sensitivity function S(u)are supposed to satisfy D(u)≥M1(u+1)^−αand S(u)≤M2(u+1)^β,respectively,where M1,M2>0 and α,β∈R.Moreover,the logistic source f(u)is supposed to satisfy f(u)≤a−μu^γ with μ>0,γ≥1,and a≥0.Asα+2β<γ−1+2γ/n,we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.

DYNAMICS FOR A CHEMOTAXIS MODEL WITH GENERAL LOGISTIC DAMPING AND SIGNAL DEPENDENT MOTILITY

In this paper,we consider the fully parabolic Chemotaxis system with the general logistic source{ut=Δ(γ(v)u)+λu-μu^(k),x∈Ω,t>0,vt=△v+wz,x∈Ω,t>0,wt=-wz,x∈Ω,t>0,zt=△z-z+u,x∈Ω,t>0 whereΩ⊂ℝn(n≥1)is a smooth and bounded domain,λ≥0,μ≥0,κ>1,and the motility function satisfies thatγ(v)∈C3([0,∞)),γ(v)>0,γ′(v)≤0 for all v≥0.Considering the Neumann boundary condition,we obtain the global boundedness of solutions if one of the following conditions holds:(i)λ=μ=0,1≤nλ3;(ii)λ>0,μ>0,combined withκ>1,1≤n≤3 or k>n+2/4,,n>3.Moreover,we prove that the solution (u, v, w, z) exponentially converges to the constant steady state ((λ/μ)1/k-1,∫Ωv0dx+∫Ωw0dx/|Ω|,0,(λ/μ)1/k-1).

Global Dynamics of a Quasilinear Chemotaxis System with Indirect Signal Production and Generalized Logistic Source

This paper considers the following quasilinear chemo-taxis system with indirect signal production and generalized logistic source{u_(t)=▽·(▽(u)▽u)-▽·(S(u)▽v)+μ(u-u^(Y)),x∈Ω,t>0 O=△v-v+w,x∈Ω,t>0 w_(t)=-δw+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domainΩ■R^(n)(n≥1),where the parametersμ,δ>0 andγ>1,the functions D(u)and S(u)are supposed to be smooth fulfilling D(u)≥C_(D)(u+1)^(-a)and S(u)≤C_(S)u(u+1)^(β-1)for all u≥0 with C D,α,β∈R and.It is proved that the corresponding initial-boundary value problem possesses a global bounded classical solution ifα+2β<γ-1.Moreover,ifμis suitably large,the asymptotic behavior and convergence rates are also been considered.

Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper,we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source{ut=■·(D(u)■u)-■·(S(u)■v)+b-μur,x∈Ω,t>0,vt=Δv+a1v+b1w,x∈Ω,t>0,wt=Δw-a2w+b2u,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domainΩ■R^(N)(N≥1),where b≥0,γ≥1,ai≥1,μ,bi>0(i=1,2,D,S∈C^(2)([0,∞))fulfilling D(S)≥a0(s+1)^(-a),0≤S(s)≤b0(s+1)^(β)for all s≥0,where a0,b0>0 and a,β≤R are constants.The pur purpose of this paper is to prove that if b≥0 andμ>0 sufficiently large,the globally bounded solution(u,v,w)with nonnegative initial data(u0,v0,w0)satisfies||u(·,t)-(b/μ)^(1/γ)L∞(Ω)+||V(·,t)+b1b2/a1a2(b/μ)^(1/γ)||L∞(Ω)+||w(·,t)+b2/a2(b/μ)^(1/γ)||L∞(Ω)→0ast→∞.

On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source

We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard(CH)equation in this paper.It is a well-known fact that the maximum principle does not hold for the CH equation.Therefore,a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time.To overcome this drawback,we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow.We consider the temporal profiles of numerical results in the one-,two-,and three-dimensional spaces to examine the effect of extra mass source.Numerical solutions are obtained using a finite difference multigrid solver.Moreover,we perform numerical tests for tumor growth simulation,which is a typical application of generalized CH equations in biology.We apply the proposed cut-off logistic source term and have good results.