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Volume 38, Issue 3
Threshold Solutions for Nonlocal Reaction Diffusion Equations

He Zhang, Yong Li & Xue Yang

DOI: 10.4208/cmr.2022-0003

Commun. Math. Res., 38 (2022), pp. 389-421.

Published online: 2022-08

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  • Abstract

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theoretical results.

  • Keywords

Nonlocal reaction diffusion equation, asymptotic behaviors, threshold solution, sharp transition.

  • AMS Subject Headings

35K57, 35B40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-389, author = {Zhang , HeLi , Yong and Yang , Xue}, title = {Threshold Solutions for Nonlocal Reaction Diffusion Equations}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {3}, pages = {389--421}, abstract = {

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theoretical results.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0003}, url = {http://global-sci.org/intro/article_detail/cmr/20962.html} }
TY - JOUR T1 - Threshold Solutions for Nonlocal Reaction Diffusion Equations AU - Zhang , He AU - Li , Yong AU - Yang , Xue JO - Communications in Mathematical Research VL - 3 SP - 389 EP - 421 PY - 2022 DA - 2022/08 SN - 38 DO - http://doi.org/10.4208/cmr.2022-0003 UR - https://global-sci.org/intro/article_detail/cmr/20962.html KW - Nonlocal reaction diffusion equation, asymptotic behaviors, threshold solution, sharp transition. AB -

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theoretical results.

Zhang , HeLi , Yong and Yang , Xue. (2022). Threshold Solutions for Nonlocal Reaction Diffusion Equations. Communications in Mathematical Research . 38 (3). 389-421. doi:10.4208/cmr.2022-0003
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