# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021186
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## Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation

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

* Corresponding author

Received  March 2021 Early access July 2021

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

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.

Citation: Pengchao Lai, Qi Li. Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021186
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