The authors study the localization of solutions to the Cauchy problem for a doubly degenerate parabolic equation with a strongly nonlinear source \[ u_t=\text{div\,}(|\nabla u^m|^{p-2}\nabla u^l)+u^q,\quad (x,t)\in \mathbb R^N\times (0,T), \] where \(N \geq 1,\) \(p > 2\) and \(m, l, q > 1.\) In the case when \(q > l + m(p - 2),\) it is proved that the solution \(u(x, t)\) has strict localization if the initial data \(u_0(x)\) has a compact support, while \(u(x, t)\) has the property of effective localization if the initial data possesses radially symmetric decay. Moreover, when \(1 < q < l + m(p - 2),\) it turns out that the solution of the Cauchy problem blows up at any point of \(\mathbb R^N\) for arbitrary initial data with compact support.