The uniqueness of the solution to the diffusion equation with a damping term

ABSTRACT

Consider a non-linear diffusion equation with a damping term. If the diffusion coefficient is positive, then the solutions are not unique generally. However, if the diffusion coefficient degenerates, the situation may change. In this paper, not only the existence of the weak solution is established, but also the uniqueness of the weak solutions is proved, even the boundary value condition is not imposed. The conclusions imply that, on the boundary, the degeneracy of diffusion coefficient can eliminate the action from the damping term.

KEYWORDS:

• Diffusion equation
• boundary value condition
• damping term
• uniqueness

AMS SUBJECT CLASSIFICATIONS:

• 35L65
• 35L85
• 35R35

Acknowledgements

I would like to express my very sincerely thanks to the reviewers for their valuable amendments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The paper is supported by Natural Science Foundation of Fujian province [grant number 2015J01592], supported by Science Foundation of Xiamen University of Technology, China.