The initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space

Abstract

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions.

Keywords:

• A weighted Sobolev space
• Faedo–Galerkin method
• Poincaré-type operator
• initial-nonlinear nonlocal conditions
• a parabolic system

AMS subject classification::

• 34B60
• 35K55
• 35Q72
• 80A30

Acknowledgments

The authors wish to express their sincere thanks to the editor and the referees for the valuable comments and remarks for the improvement of the paper. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2020-18-01.

Disclosure statement

No potential conflict of interest was reported by the author(s).