  While much literature on chemotaxis systems focuses on bounded domains, this
paper emphasizes the global existence of classical solutions for three primary
chemotaxis systems with a logistic source on $\mathbb{R}^n$. We present a
unified proof demonstrating global existence of solutions can be deduced from
their locally uniform boundedness in $L^p(\mathbb{R}^n)$ for some
$p>\max\{1,\frac{n}{2}\}$. We then provide sufficient conditions for the global
existence and boundedness of classical solutions. Notably, our findings even
improve several existing results for bounded domains.
