A reaction–diffusion system governed by nonsmooth semipermeability problem

Abstract

Recently, in [Tang GJ, Cen JX, Nguyen VT, et al. Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence. J Fixed Point Appl. 2020; DOI: 10.1007/s11784-020-00814-4],we studied a comprehensive system called differential variational-hemivar-iational inequality (DVHVI, for short) which is composed of a nonlinear evolution equation and a time-dependent variational-hemivariational inequality in Banach spaces. We have proved the existence, uniqueness, and stability of the solution in mild sense, as well as a surprising convergence result for DVHVI. However, to illustrate the applicability of those theoretical results in Tang et al., the present paper is devoted to explore a coupled dynamic system which is formulated by a nonlinear reaction–diffusion equation described by a time-dependent nonsmooth semipermeability problem.

Keywords:

• Reaction–diffusion equation
• nonsmooth semipermeability problem
• differential variational-hemivariational inequality
• $C_0$-semigroup
• Clarke's subgradient

2010 Mathematics Subject Classifications:

• 35K55
• 35K57
• 47J20
• 49J40
• 58E35
• 35K90
• 46Txx

Disclosure statement

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

Additional information

Funding

This project has received funding from the NNSF of China Grant Nos. 12001478,12026255,12026256, and the European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It is also supported by Natural Science Foundation of Guangxi Grant No. 2020GXNSFBA297137, and the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0,440328/PnH2/2019. The second author was also supported by National Natural Science Foundation of China Grant No. 11961006 and Natural Science Foundation of Guangxi (NSF of Guangxi) No. 2020GXNSFAA159100.