June  2022, 27(6): 3313-3323. doi: 10.3934/dcdsb.2021186

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 Published  June 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 3313-3323. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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