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In this paper, a nonlinear porous medium equation under a new condition onnonlinearity is considered in a bounded domain. Firstly, we present the blow-upof the positive solution to the considered problem for the negative initialenergy. By introducing a family of potential wells, we establish the invariantsets and vacuum isolating solutions of nonlinear porous medium equation. Forthe subcritical and critical initial energy cases, we obtain a globalexistence, asymptotic behavior and blow-up phenomena in a finite time of thepositive solution to the nonlinear porous medium equation.

Global existence and blow-up of solutions to the nonlinear porous medium equation