Blowup and global existence of a free boundary problem with weak spatial source

Abstract

This paper deals with the blowup and global existence of the solutions for a nonlinear reaction–diffusion equation with free boundaries and very weak spatial sources. For such a problem, we first derive some sufficient conditions to finite time blowup of the solution. Secondly, the existence of global vanishing solutions is proven for a family of sufficiently small initial values. Finally, a sharp threshold trichotomy result is obtained in term of the size of the initial value, by which the blowup solution, the global vanishing solution, and, in particular, the global transition solution are distinguished.

COMMUNICATED BY:

• Dinghua Xu

Keywords:

• Nonlinear reaction–diffusion equation
• free boundary
• blowup
• global solution
• sharp threshold trichotomy

AMS Subject Classification (2010):

• 35K57
• 35R35
• 35B33
• 35B40

Acknowledgments

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions, which have expressly promoted the original manuscript. Moreover, the authors would like to still express the special thanks to Professor Mingxin Wang for his positive suggestion.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 11701243], the HongLiu first-class disciplines Development Program of Lanzhou University of Technology (J. Wang), and the Fundamental Research Funds for the Central Universities, China University of Geosciences [grant number 162301182740] (J.-B. Wang).