The global solution and blowup of a equation modeled from the water wave problem with critical growth

This study is led by Dr. Zhong Tan (School of Mathematical Sciences, Xiamen University) and Dr. Yiying Wang (School of Mathematical Sciences, Xiamen University). They overcome the difficulty cased by the nonlocal opeartor and the lack of compactness, and study the suitability and asymptotic behavior of global solution of water wave problem.

Firstly, they proved when the solution possess negative energy initial value, the solution will blow up in finite time. And this is proved by the concavity method.

Based on the above result, they then proved that solutions with low-energy initial value not only exist globally but also decay exponentially over time. In contrast, if the initial configuration exceeds this energy threshold, the solution will blow up in finite time.

Secondly, the group proved that the regularity for solution with low-energy initial value. The team achieved this by adeptly applying the Morse iteration and bootstrap technique

, a powerful method that progressively enhances the regularity of the solution. used a Morse the regularity of the global solution with low-energy initial value.

A significant challenge in studying the long-term behavior of such equation is that when the nonlinear term is with the the critical growth, the Sobolev trace embedding loses its compactness. To overcome this, the team employed the concentration-compactness principle, which was first introduced by P.L.Lions. This allowed them to prove that the global solutions will concentrate around a finite number of specific profiles, each related to a stationary solution of the related stable equation.

See the article:

The global solution and blowup of an equation modeled from the water wave problem with critical growth

https://doi.org/10.1007/s10473-025-0608-6