In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution f(t, v) is bounded pointwise from above by \(C_{f_0}\langle t \rangle ^3\) and establish that the cooling time is infinite ( \( T_c = +\infty \) ) under the condition \( f_0 \in L^1_2 \cap L^{\infty }_{s} \) for \( s > 2 \) . Away from zero velocity, we further prove that \( f(t,v)\le C_{f_0, |v|} \langle t \rangle \) for \(v \ne 0\) at any time \( t > 0 \) . This time-dependent pointwise upper bound is natural in the cooling process, as we expect the density near \( v = 0 \) to grow rapidly. We also establish an upper bound that depends on the coefficient of normal restitution constant, \(\alpha \in (0,1]\) . This upper bound becomes constant when \(\alpha = 1\) , restoring the known upper bound for elastic collisions [8]. Consequently, through these results, we obtain Maxwellian upper bounds on the solutions at each time.
