Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds

Abstract

In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

Keywords

• global existence
• finite-time blowup
• nonlinear parabolic equation
• singular manifolds
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