The paper studies the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system in higher dimensions (n ≥ 3). It proves that for any prescribed mass m > 0, there exist radially symmetric initial data (u₀, v₀) such that the corresponding solution blows up in finite time. The study shows that within the space of radial functions, the set of initial data leading to finite-time blow-up is large in an appropriate sense, including being dense in certain function spaces. The key result is that the energy inequality for the system can be used to derive a sufficient condition for finite-time blow-up. The paper also provides an explicit blow-up criterion and demonstrates that the set of initial data enforcing finite-time blow-up is dense in L^p(Ω) × W^{1,2}(Ω) for p ∈ (1, 2n/(n+2)). The results highlight the complex behavior of the Keller-Segel system in higher dimensions, showing that even with radial symmetry and bounded mass, solutions can blow up in finite time under certain conditions.The paper studies the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system in higher dimensions (n ≥ 3). It proves that for any prescribed mass m > 0, there exist radially symmetric initial data (u₀, v₀) such that the corresponding solution blows up in finite time. The study shows that within the space of radial functions, the set of initial data leading to finite-time blow-up is large in an appropriate sense, including being dense in certain function spaces. The key result is that the energy inequality for the system can be used to derive a sufficient condition for finite-time blow-up. The paper also provides an explicit blow-up criterion and demonstrates that the set of initial data enforcing finite-time blow-up is dense in L^p(Ω) × W^{1,2}(Ω) for p ∈ (1, 2n/(n+2)). The results highlight the complex behavior of the Keller-Segel system in higher dimensions, showing that even with radial symmetry and bounded mass, solutions can blow up in finite time under certain conditions.