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Existence and stability of steady-state solutions of reaction–diffusion equations with nonlocal delay effect

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Abstract

A general reaction–diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady-state solutions are proved via studying an equivalent reaction–diffusion system without nonlocal and delay structure and applying local and global bifurcation theory. The global structure of the set of steady states is characterized according to type of nonlinearities and diffusion coefficient. Our general results are applied to diffusive logistic growth models and Nicholson’s blowflies-type models.

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Acknowledgements

This work was completed when the first author visited William & Mary in 2015–2016, and she would like to thank W&M for warm hospitality.

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Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, People’s Republic of China

    Wenjie Zuo

  2. Department of Mathematics, William & Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

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  1. Wenjie Zuo
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Correspondence to Junping Shi.

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Partially supported by the NSFC of China (No. 11671236), the Natural Science Foundation of Shandong Province of China (No. ZR2019MA006), the Fundamental Research Funds for the Central Universities (No. 19CX02055A), China Scholarship Council and US-NSF grants DMS-1715651 and DMS-1853598.

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Zuo, W., Shi, J. Existence and stability of steady-state solutions of reaction–diffusion equations with nonlocal delay effect. Z. Angew. Math. Phys. 72, 43 (2021). https://doi.org/10.1007/s00033-021-01474-1

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