Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent

Abstract

In this paper, we study a class of quasilinear parabolic equations involving -Laplacian operator and nonlocal term in a bounded domain, which arises from biomathematics or nonstationary fluids. Under appropriate assumptions, we obtain the local existence of weak solutions by applying Galerkin’s approximation method.

Keywords:

• parabolic equation of Kirchhoff type
• variable exponent space
• Galerkin approximation
• local existence

AMS Subject Classifications:

• 35K20
• 35K55
• 35D30
• 35Q92

Acknowledgements

The authors would like to thank an anonymous referee for many helpful comments and suggestions on the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11371110]; the Scientific Research Foundation of Civil University of China [grant number 2014QD04X].