This paper deals with the semilinear Cauchy problem for the equation \(\boxed {\phantom \cdot} u=V(x)u^5\) in the 3-dimensional case with \(V(x)\geq 0\) and \(V\in C^2\). Global existence results in \(t\geq 0\) are proved in several cases, namely \(V(x)=|x-\overline{x}|^\alpha\), \(\alpha=0\), \(\alpha\geq 2\); \(V=x_1^{2n_1}+x_2^{2n_2}+x_3^{2n_3}\), \(n_1, n_2, n_3\geq 2\) being integers and under the additional restriction the initial energy to be sufficiently small.