Blow-up with mass concentration for the long-wave unstable thin-film equation

Abstract

For a family of long-wave unstable thin-film equations, we prove existence of non-negative weak solutions blowing-up in a finite time. Specifically, building these solutions from initial data with negative energy, we show that their -norms go to infinity as . In addition, using the Bourgain’s type approach, we obtain qualitative information about the blow-up and prove mass concentration phenomenon.

Keywords:

• thin-film equation
• long-wave instability
• blow-up
• mass concentration

AMS Subject Classifications:

• 35K55
• 35K35
• 35K65
• 76A20
• 76D08

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by a grant from the Simons Foundation [grant number 275088] to Marina Chugunova.