The paper investigates the finite-time blow-up behavior of solutions to the parabolic-parabolic Keller-Segel system in higher-dimensional spaces. The authors prove that for any prescribed mass \( m > 0 \), there exist radially symmetric positive initial data \((u_0, v_0)\) such that the corresponding solution blows up in finite time. They also provide an explicit blow-up criterion and show that the set of such initial data is dense in the space of all radial functions with respect to the topology of \( L^p(\Omega) \times W^{1,2}(\Omega) \) for any \( p \in (1, \frac{2n}{n+2}) \). This indicates that finite-time blow-up is a common phenomenon for this system in three or more dimensions. The main results are derived using energy estimates and a detailed analysis of the dissipated quantity, leading to a blow-up criterion that is essentially explicit. The paper also includes a proof of the density property of the set of initial data enforcing finite-time blow-up, which is a significant contribution to the understanding of this system.The paper investigates the finite-time blow-up behavior of solutions to the parabolic-parabolic Keller-Segel system in higher-dimensional spaces. The authors prove that for any prescribed mass \( m > 0 \), there exist radially symmetric positive initial data \((u_0, v_0)\) such that the corresponding solution blows up in finite time. They also provide an explicit blow-up criterion and show that the set of such initial data is dense in the space of all radial functions with respect to the topology of \( L^p(\Omega) \times W^{1,2}(\Omega) \) for any \( p \in (1, \frac{2n}{n+2}) \). This indicates that finite-time blow-up is a common phenomenon for this system in three or more dimensions. The main results are derived using energy estimates and a detailed analysis of the dissipated quantity, leading to a blow-up criterion that is essentially explicit. The paper also includes a proof of the density property of the set of initial data enforcing finite-time blow-up, which is a significant contribution to the understanding of this system.