Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition

ABSTRACT

In this paper, we consider the following nonlocal autonomous evolution equation in a bounded domain Ω in ${\mathbb{R}}^N$: ${\mathrm{\partial }}_{t}u(x,t)=-h\left(x\right)u(x,t)+g({\int }_{\mathrm{\Omega }}J(x,y)u(y,t)\mathrm{d}y)+f(x,u(x,t\left)\right),$ where $h\in W^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, $g:\mathbb{R}\to \mathbb{R}$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ are continuously differentiable function, and J is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any $x,y\in {\mathbb{R}}^N$. Under additional suitable assumptions on f and g, we study the asymptotic dynamics of the initial value problem associated to this equation in a suitable phase spaces. More precisely, we prove the existence, and upper semicontinuity of compact global attractors with respect to kernel J.

KEYWORDS:

• Asymptotic dynamics
• nonlocal equation
• global attractor
• upper semicontinuity

2010 MATHEMATICAL SUBJECT CLASSIFICATIONS:

• 45J05
• 45M05
• 34D45

Acknowledgements

The authors are thankful to the anonymous referee for his or her valuable corrections and comments on early version of this work that helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

S. Sastre-Gomez was partially supported by Propesq UFPE project Qualis-A 2016 and by Spanish Ministerio de Economía y Competitividad MEC project MTM 2012-31298. S. H. da Silva's research was partially supported by CAPES/CNPq-Brazil Grant 552464/2012-2.