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
References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math, 446, Springer, Berlin, 1975, 5–49.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

[5]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312.  doi: 10.4171/JEMS/198.  Google Scholar

[6]

Y. Du and P. Poláčik, Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $\mathbb {R}^N$, Indiana Univ. Math. J., 64 (2015), 787-824.  doi: 10.1512/iumj.2015.64.5535.  Google Scholar

[7]

A. DucrotT. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), 5541-5566.  doi: 10.1090/S0002-9947-2014-06105-9.  Google Scholar

[8]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[9]

Y. Kaneko and Y. Yamada, Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions, J. Math. Anal. Appl., 465 (2018), 1159-1175.  doi: 10.1016/j.jmaa.2018.05.056.  Google Scholar

[10]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

[11]

Q. Li and B. Lou, Vanishing phenomena in fast decreasing generalized bistable equations, J. Math. Anal. Appl., 500 (2021), 125096. doi: 10.1016/j. jmaa. 2021.125096.  Google Scholar

[12]

A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263.  doi: 10.1090/S0894-0347-05-00504-7.  Google Scholar

show all references

References:
[1]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Math, 446, Springer, Berlin, 1975, 5–49.  Google Scholar

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[4]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

[5]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312.  doi: 10.4171/JEMS/198.  Google Scholar

[6]

Y. Du and P. Poláčik, Locally uniform convergence to an equilibrium for nonlinear parabolic equations on $\mathbb {R}^N$, Indiana Univ. Math. J., 64 (2015), 787-824.  doi: 10.1512/iumj.2015.64.5535.  Google Scholar

[7]

A. DucrotT. Giletti and H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Amer. Math. Soc., 366 (2014), 5541-5566.  doi: 10.1090/S0002-9947-2014-06105-9.  Google Scholar

[8]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[9]

Y. Kaneko and Y. Yamada, Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions, J. Math. Anal. Appl., 465 (2018), 1159-1175.  doi: 10.1016/j.jmaa.2018.05.056.  Google Scholar

[10]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

[11]

Q. Li and B. Lou, Vanishing phenomena in fast decreasing generalized bistable equations, J. Math. Anal. Appl., 500 (2021), 125096. doi: 10.1016/j. jmaa. 2021.125096.  Google Scholar

[12]

A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), 251-263.  doi: 10.1090/S0894-0347-05-00504-7.  Google Scholar

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