Long-time behavior of global solutions to the 1-equivariant harmonic map heat flow and the energy critical heat equations

Profile decomposition is an intriguing phenomenon for the energy critical equations, like harmonic map heat flow in $R^2$ and energy critical heat equations. It is a natural question to ask which configuration of bubbles is possible in the decomposition. Inspired by the formal analysis of Fila-King for the critical heat equations, we construct global solutions for the harmonic map heat flow in the 1-equivariant class and the energy critical heat equations. This is also related to the behavior of threshold solutions. This talk is based on several work joint with Juncheng Wei, Yifu Zhou, and Zaizheng Li.