Asymptotic behavior to a chemotaxis consumption system with singular sensitivity

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We consider a chemotaxis consumption system with singular sensitivity , vt=εΔv−uv in a bounded domain with χ,α,ε>0. The global existence of classical solutions is obtained with n=1. Moreover, for any global classical solution (u,v) to the case of n,α≥1, it is shown that v converges to 0 in the L∞‐norm as t→∞ with the decay rate established whenever ε∈(ε0,1) with .