Semilinear parabolic equations in Herz spaces

Abstract

In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=G\left(u\right)$ with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces.

Keywords:

• Herz spaces
• parabolic equations
• maximal function

2020 Mathematics Subject Classifications:

• 35K55
• 42B35
• 46E30

Acknowledgments

We would like to thank the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is founded by the General Direction of Higher Education and Training [grant number C00L03UN280120220004] and by The General Directorate of Scientific Research and Technological Development.