  In this study, we examine a double nonlinear porous medium equation subject
to a novel nonlinearity condition within a bounded domain. First, we introduce
the blow-up solution for the problem under consideration for the negative
initial energy. By introducing a set of potential wells, we construct invariant
sets of solutions for the double nonlinear porous medium equation. For
subcritical and critical initial energy scenarios, we derive the global
existence and asymptotic behavior of weak solutions, as well as blow-up
phenomena occurring within a finite time for the positive solution to the
double nonlinear porous medium equation.
