Global existence and boundedness of solutions to a fully parabolic chemotaxis system with indirect signal production in ${\mathbb{R}}^4$

Global existence and boundedness of solutions to the Cauchy problem for the four dimensional fully parabolic chemotaxis system with indirect signal production are studied. We prove that solutions with initial mass below $(8\pi {)}^2$ exist globally in time. This value $(8\pi {)}^2$ is known as the four dimensional threshold value of the initial mass determining whether blow-up of solutions occurs or not. Furthermore, some condition on the initial mass guaranteeing that the solution remains uniformly bounded is also obtained.