间接信号吸收的趋化模型解的全局有界性

Global boundedness of solution for the chemotaxis model with indirect signal absorption

考虑具有间接信号吸收的高维拟线性趋化增长模型ut=∇·(D(u)∇u)-∇·(S(u)∇v)+μ(u-u2),(x,t)∈Ω×(0,T),vt=Δv-vw,(x,t)∈Ω×(0,T),wt=-w+u,(x,t)∈Ω×(0,T),􀆟u􀆟ν=􀆟v􀆟ν=0,(x,t)∈Ω×(0,T),u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω,其中Ω⊂Rn(n≥3)是一个有界区域且具有光滑边界,μ>0,非线性扩散系数D(u)和趋化敏感系数S(u)满足D(u)≥u^(m-1),S(u)≤u^(q-1)且D(·),S(·)∈C^(1+l)[0,∞),l>0.利用能量方法和半群理论证明了:当m>max{1,2q-3}时,该生物趋化模型的解全局有界.

The high-dimensional quasilinear chemotaxis-growth model with indirect signal absorptionut=∇·(D(u)∇u)-∇·(S(u)∇v)+μ(u-u2),(x,t)∈Ω×(0,T),vt=Δv-vw,(x,t)∈Ω×(0,T),wt=-w+u,(x,t)∈Ω×(0,T),􀆟u􀆟ν=􀆟v􀆟ν=0,(x,t)∈􀆟Ω×(0,T),u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω on a bounded domainΩ⊂Rn(n≥3)with smooth boundary􀆟Ωis discussed,whereμis positive constant,the nonlinear diffusivity D(u)and chemosensitivity S(u)are supposed to satisfy D(u)≥u^(m-1),S(u)≤u^(q-1) and D(·),S(·)∈C^(1+l)[0,∞),l>0,m,μ,q,l are paramaters.By aid of the energy method and the semigroup theory,it is proved that the solution of the biological chemotaxis model is globally bounded under the condition of m>max{1,2q-3}.