This paper concerns the global existence and decay estimates of classical solutions to a chemotaxis-consumption system on \({\mathbb {R}}^N\) , for all \( N\ge 1 \) . First, we construct the local classical solution, and then by some a priori estimates, we extend the local solution step by step to all \( t > 0 \) provided that \( \Vert u_0\Vert _{L^1({\mathbb {R}}^N)}+\Vert \nabla v_0\Vert _{L^1({\mathbb {R}}^N)} \) is sufficiently small. Finally, based on the Fourier splitting method, we build the temporal decay estimates for the global solutions.
