Abstract
This paper is concerned with existence and stability of V-shaped traveling fronts for a class of nonlocal dispersal equations with unbalanced bistable nonlinearity. The main tool is sub- and supersolution technique combined with a comparison principle.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bates, P.W., Fife, P.C., Ren, X., Wang, X.: Traveling waves in a convolution model for phase transitions. Arch. Ration. Mech. Anal. 138, 105–136 (1997)
Bonnet, A., Hamel, F.: Existence of non-planar solutions of a simple model of premixed Bunsen flames. SIAM J. Math. Anal. 31, 80–118 (1999)
Brazhnik, P.K., Tyson, J.J.: On traveling wave solutions of Fisher’s equation in two spatial dimensions. SIAM J. Appl. Math. 60, 371–391 (1999)
Bu, Z.-H., Ma, L., Wang, Z.-C.: Conical traveling fronts of combustion equations in \(\mathbb{R}^3\). Appl. Math. Lett. 108, 106509 (2020)
Bu, Z.-H., Wang, Z.-C.: Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations. Discrete Contin. Dyn. Syst. 38, 2251–2286 (2018)
Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)
Chan, H., Wei, J.: Traveling wave solutions for bistable fractional Allen–Cahn equations with a pyramidal front. J. Differ. Equ. 262, 4567–4609 (2017)
Chen, X.: Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)
Coville, J.: Equations de reaction diffusion non-locale. Mathématiques. Universitsé Pierre et Marie Curie-Paris VI, Franxcais (2003)
Coville, J., Dupaigne, L.: Travelling fronts in integrodifferential equations. C. R. Acad. Sci. Paris, Ser. I 337, 25–30 (2003)
Hamel, F., Nadirashvili, N.: Travelling fronts and entire solutions of the Fisher-KPP equation in \(\mathbb{R}^{N}\). Arch. Ration. Mech. Anal. 157, 91–163 (2001)
Hamel, F., Monneau, R., Roquejoffre, J.-M.: Existence and qualitative properties of multidimensional conical bistable fronts. Discrete Contin. Dyn. Syst. 13, 1069–1096 (2005)
Li, W.-T., Sun, Y.-J., Wang, Z.-C.: Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal. Real World Appl. 11, 2302–2313 (2010)
Li, W.-T., Niu, H.-T., Wang, Z.-C.: Nonplanar traveling fronts for nonlocal dispersal equations with bistable nonlinearity. Differ. Integral Equ. 34, 265–294 (2021)
Ni, W.-M., Taniguchi, M.: Traveling fronts of pyramidal shapes in competition–diffusion systems. Netw. Heterog. Media 8, 379–395 (2013)
Ninomiya, H., Taniguchi, M.: Existence and global stability of traveling curved fronts in the Allen–Cahn equations. J. Differ. Equ. 213, 204–233 (2005)
Ninomiya, H., Taniguchi, M.: Global stability of traveling curved fronts in the Allen–Cahn equations. Discrete Contin. Dyn. Syst. 15, 819–832 (2006)
Niu, H.-T., Wang, Z.-C., Bu, Z.-H.: Curved fronts in the Belousov–Zhabotinskii reaction-diffusion systems in \(\mathbb{R}^{2}\). J. Differ. Equ. 264, 5758–5801 (2018)
Schumacher, K.: Travelling-front solutions for integro-differential equations. I. J. Reine Angew. Math. 316, 54–70 (1980)
Taniguchi, M.: Traveling fronts of pyramidal shapes in the Allen–Cahn equations. SIAM J. Math. Anal. 39, 319–344 (2007)
Taniguchi, M.: The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen–Cahn equations. J. Differ. Equ. 246, 2103–2130 (2009)
Taniguchi, M.: Multi-dimensional traveling fronts in bistable reaction–diffusion equations. Discrete Contin. Dyn. Syst. 32, 1011–1046 (2012)
Taniguchi, M.: An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen–Cahn equation. SIAM J. Math. Anal. 47, 455–476 (2015)
Taniguchi, M.: Convex compact sets in \(\mathbb{R}^{N-1}\) give traveling fronts of cooperation–diffusion systems in \(\mathbb{R}^{N}\). J. Differ. Equ. 260, 4301–4338 (2016)
Wang, Z.-C.: Traveling curved fronts in monotone bistable systems. Discrete Contin. Dyn. Syst. 32, 2339–2374 (2012)
Wang, Z.-C., Bu, Z.-H.: Nonplanar traveling fronts in reaction–diffusion equations with combustion and degenerate Fisher-KPP nonlinearities. J. Differ. Equ. 260, 6405–6450 (2016)
Wang, Z.-C., Li, W.-T., Ruan, S.: Existence, uniqueness and stability of pyramidal traveling fronts in reaction–diffusion systems. Sci. China 59, 1869–1908 (2016)
Wang, Z.-C., Niu, H.-L., Ruan, S.: On the existence of axisymmetric traveling fronts in Lotka–Volterra competition–diffusion systems in \(\mathbb{R}^{3}\). Discrete Contin. Dyn. Syst. Ser. B 22, 1111–1144 (2017)
Zhang, G.-B., Li, W.-T., Wang, Z.-C.: Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. J. Differ. Equ. 252, 5096–5124 (2012)
Acknowledgements
The work is supported by NNSF of China (11901330).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Niu, HT. Existence and stability of traveling curved fronts for nonlocal dispersal equations with bistable nonlinearity. Z. Angew. Math. Phys. 73, 90 (2022). https://doi.org/10.1007/s00033-022-01734-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-022-01734-8
Keywords
- Unbalanced bistable nonlinearity
- Nonlocal dispersal
- V-shaped traveling fronts
- Super- and subsolutions
- Stability