Pullback attractors in the weighted space for multi-valued process generated by the non-autonomous nonclassical diffusion equations with unbounded delays without uniqueness of solutions

ABSTRACT

In this article, we consider the asymptotic behavior of the following non-autonomous nonclassical diffusion equations with infinite delays $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\mathrm{\Delta }\frac{\mathrm{\partial }u}{\mathrm{\partial }t}-\mathrm{\Delta }u=f(t,u(t-\rho \left(t\right),x\left)\right)+{\int }_{-\mathrm{\infty }}^{0}G(t,s,u(t+s,x\left)\right)\mathrm{d}s+g(x,t).$ Under suitable conditions on f and G, we prove the existence of pullback attractors in the weighted space $C_{\gamma ,H_0^1\left(\mathrm{\Omega }\right)}$ for multi-valued process generated by non-autonomous nonclassical diffusion equations with unbounded delays without uniqueness of solutions while the external force $g(x,t)\in L_{C^{\ast }}^2(\mathbb{R},L^2(\mathrm{\Omega }\left)\right)$.

COMMUNICATED BY:

• Meir Shillor

KEYWORDS:

• Nonclassical diffusion equations
• pullback attractors
• unbounded delays

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B41
• 35K57

Acknowledgments

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. They also thank the editors for their kind help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

F. Zhang was supported by the 2018 research funding of higher education of Gansu province project [2018B-075]. L. Bai was supported by the 2017 research funding of higher education of Gansu province project [2017A-185].