Asymptotic behavior of solutions for time-delayed nonlocal dispersion equations with Dirichlet boundary

Abstract

This paper is concerned with the asymptotic behavior of the solutions for Nicholson's blowflies equation with nonlocal dispersion subjected to Dirichlet boundary condition. We first prove the existence and uniqueness of the solution for the initial boundary value problem and its non-trivial steady state. Then we give a threshold result on global stability of equilibria: when $D{\lambda }_1+\delta >p$, the solution time-exponentially converges to the constant equilibrium 0 for any large initial data; when $D{\lambda }_1+\delta <p$, the solution time-asymptotically converges to its positive steady-state $\varphi (x)$ for any large initial data, once $1<\frac{p}{\delta }\le e$, where D>0 is the diffusion coefficient, $\delta >0$ is the death rate, p>0 is the birth rate and ${\lambda }_1>0$ is the principal eigenvalue for the nonlocal characteristic equation. The adopted approach is the energy method and the monotonic technique.

Keywords:

• Nicholson's blowflies equation
• nonlocal dispersion
• time delay
• global stability
• steady-state solutions

2010 Mathematics Subject Classifications:

• 35K57
• 35B40
• 35C07
• 35K58
• 92D25

Acknowledgments

The research by YCJ was supported in part by the NSFC (No. 11571066). The research by KJZ was supported in part by the NSFC (No. 11371082 and No. 11771071) and the Fundamental Research Funds for the Central Universities (No. 111065201).

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [11371082,11771071,11571066].