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In this paper, we consider the attraction‐repulsion parabolic‐parabolic‐parabolic model with nonlinear diffusion ut=∇·(D(u)∇u)−∇·(uS1(x,u,v)∇v)+∇·(uS2(x,u,w)∇w)+ξu−μu2,x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0, in a smooth bounded domain Ω⊂ℝn(n≥2) , where α, β, γ, δ, μ are given positive parameters and ξ ≥ 0. The function… Click to show full abstract
In this paper, we consider the attraction‐repulsion parabolic‐parabolic‐parabolic model with nonlinear diffusion ut=∇·(D(u)∇u)−∇·(uS1(x,u,v)∇v)+∇·(uS2(x,u,w)∇w)+ξu−μu2,x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0, in a smooth bounded domain Ω⊂ℝn(n≥2) , where α, β, γ, δ, μ are given positive parameters and ξ ≥ 0. The function D satisfies D(u) ≥ CDum − 1 for all u > 0 with CD > 0. Si(i=1,2) are given matrix‐valued functions in ℝn×n which fulfill |S1(x,u,v)|≤CS1(1+u)−α1,|S2(x,u,w)|≤CS2(1+u)−α2 with some CSi>0 and αi>0(i=1,2) . It is shown that under the conditions m > 0 and min{m+2α1,m+2α2}>2nn+2 , the corresponding initial boundary value problem possesses at least one global bounded weak solution.
Journal Title: Mathematical Methods in the Applied Sciences
Year Published: 2020
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