Studies On Properties Of Solutions To Some Chemotaxis Models And Some Related Interaction Functionals

This thesis studies the global boundedness and finite-time blow-up of solutions for several chemotaxis models,and the minimizations of some related interaction functionals.The dissertation is divided into five parts:Chapter 1 gives the background and research status of chemotaxis models and some related interaction functionals,and briefly describes our main results and innovations.In Chapter 2,we consider a chemotaxis system with nonlinear chemosensitivity and signal production subject to homogeneous Neumann boundary conditions,where BR.(0)(?) Rn with n≥1,R>0 and μ(t)=1/|BR(0)|∫BR(0)g(u)dx.We show that the model there exists a solution blowing up in finite time provided that f and g satisfy f(ξ)≥Kξk and g(ξ)≥Lξl with k+l-1>2/n.However,the system admits a global bounded classical solution when f and g satisfy f(ξ)≤Kξk and g(ξ)≤Lξl with 0<k+l-1<2/n.In Chapter 3,we investigate a chemotaxis system with rotation flux subject to homogeneous Neumann boundary conditions,where Ω is a bounded domain in R2 with smooth boundary (?)Ω,(?)and θ∈(-π/2,π/2).We show that:·Let Ω be a general bounded domain.(ⅰ)Let m>8π/cosθ,there exists nonnegative initial datum u0 satisfying ∫Ωu0dx=m,such that the corresponding nonradial solution of blows up in finite time and the blow-up point lies in Ω.(ⅱ)Let (?)Ω contain a line segment and m>4π/cosθ,there exists nonnegative initial datum u0 satisfying ∫Ωu0dx=m,such that the nonradial solution blows up in finite time and the blow-up point lies in the line segment of (?)Ω.·Let Ω=BL(0).(i)If the nonnegative radially symmetric initial datum u0 satisfies∫Ωu0dx<8π/cosθ,then the radial solution of exists globally in time.(ⅱ)If the nonnegative radially symmetric initial datum u0 satisfies ∫Ωu0dx<4π/cosθ,then the radial solution of is globally bounded.In Chapter 4,we consider the minimization of interaction functional with exogenous potential E[ρ]=1/2∫RN∫RNK(x-y)ρ(x)ρ(y)dxdy+∫RNF(x)ρ(x)dx The kernel K(x)=1/q|x|q-1/p|x|p is an endogenous potential,where q>p>-N and N≥1.The exogenous potential F is a nonnegative continuous function and satisfies F(x)→+∞ as|x|→+∞.The existence of minimizers are established based on the compactness of energyminimizing sequences and weak lower-semicontinuity of the functional.Especially,for F(x)=β|x|2(β>0)and K(x)=1/2|x|2-1/(2-N)|x|2-N(N>2),we give the explicit of global minimizer.In Chapter 5,we deal with the minimization of interaction functional with nonlinear diffusion and exogenous potential E[ρ]=∫RN1/(a-1)ρa(x)dx+1/2∫RN∫RNK(x-y)ρ(x)ρ(y)dxdy+∫RNF(x)ρ(x)dx where N≥1 and a>1,K(x)=1/q|x|q-1/p|x|p and q>p>-N.F is a nonnegative function and satisfies ‖F‖L∞(BR(0))→+∞ as R→+∞.We analyze the existence of minimizers for the functional.Furthermore,we show the explicit of global minimizer of when q=2,p=2-N and F is a power function.