A new result for global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

Abstract

We consider the following chemotaxis model{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+μ(u−u2),x∈Ω,t>0,vt−Δv=−uv,x∈Ω,t>0,(D(u)∇u−χu⋅∇v)⋅ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω on a bounded domain Ω⊂RN(N≥1), with smooth boundary ∂Ω,χ and μ are positive constants. Here, D is supposed to be smooth positive function satisfying D(u)≥CD(u+1)m−1 for all u≥0 with some CD,m>0. Besides appropriate smoothness assumptions, in this paper it is only required thatm>{1−μ2χ[1+max1≤s≤2⁡λ0(s)‖v0‖L∞(Ω)23]ifN≤2,1ifN≥3, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where λ0 is a positive constant which is corresponding to the maximal Sobolev regularity. The results of this paper extends the results of Jin (J. Differential Equations 263 (9) (2017) 5759–5772), who proved the possibility of boundness of weak solutions, in the case m>1 and N=3. To the best of our knowledge, this is the first result which gives the relationship between m and μχ that yields to the boundedness of the solution.