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

Pengchao Lai; Qi Li

doi:10.3934/dcdsb.2021186

Article Contents

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

This issue Previous Article Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem Next Article Optimal spiral-like solutions near a singular extremal in a two-input control problem

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: March 2021

Early access: July 13, 2021

Published online: July 13, 2021

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

Abstract / Introduction Full Text(HTML) Figure(4) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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. Ducrot, T. 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.