Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation

Pengchao Lai; Qi Li

doi:10.3934/dcdsb.2021186

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2022, Volume 27, Issue 6: 3313-3323. Doi: 10.3934/dcdsb.2021186

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Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation

Pengchao Lai and
Qi Li^,

Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China

* Corresponding author
* Corresponding author

Received: February 28, 2021

Published: June 2022

This research was partly supported by the NSFC (No. 12071299)

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Abstract

In this paper we consider the reaction diffusion equation $ u_t = u_{xx} + f(u) $ with bistable-bistable type of nonlinearities, that is, $ f $ has five nonnegative zeros: $ 0<\alpha_1 <\alpha_2<\alpha_3 <\alpha_4 $, and it is of bistable type on $ [0,\alpha_2] $ and $ [\alpha_2, \alpha_4] $. We study the asymptotic behavior for the solutions under different conditions for $ k_4 : = \int_0^{\alpha_4} f(s) ds $ and $ k_2: = \int_0^{\alpha_2} f(s) ds $. In case $ k_4 > k_2 > 0 $ (resp. $ k_4 > k_2 = 0 $, $ k_2 < 0 < k_4 $, $ k_2 < k_4 = 0 $), we find 5 (resp. 3, 3, 1) possible choices for the $ \omega $-limit of the solution.

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Mathematics Subject Classification: Primary: 35K15, 35K55;Secondary: 35B40.

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