This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.
Citation: Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity[J]. Electronic Research Archive, 2021, 29(3): 2269-2291. doi: 10.3934/era.2020116
This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.
[1] | P. W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13–52. |
[2] | Traveling waves in a convolution model for phase transitions. Arch. Rational Mech. Anal. (1997) 138: 105-136. |
[3] | Generalized transition waves and their properties. Comm. Pure Appl. Math. (2012) 65: 592-648. |
[4] | Bistable traveling waves around an obstacle. Comm. Pure Appl. Math. (2009) 62: 729-788. |
[5] | A non-local bistable reaction-diffusion equaiton with a gap. Discrete Contin. Dyn. Syst. (2017) 37: 685-723. |
[6] | Uniqueness of travelling waves for nonlocal monostable equations. Proc. Amer. Math. Soc. (2004) 132: 2433-2439. |
[7] | Existence, uniqueness and asymptotic stablility of traveling waves in nonlocal evolution equations. Adv. Differential Equations (1997) 2: 125-160. |
[8] | Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differential Equations (2008) 244: 3080-3118. |
[9] | On a nonlocal reaction diffusion equation arising in population dynamics. Proc. Roy. Soc. Edinburgh Sect. (2007) 137: 727-755. |
[10] | A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities. J. Differential Equations (2018) 265: 804-829. |
[11] | A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section. Nonlinear Anal. (2018) 172: 99-114. |
[12] | P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin (2003), 153–191. |
[13] | Entire solutions of the KPP equation. Comm. Pure Appl. Math. (1999) 52: 1255-1276. |
[14] | Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$. Arch. Ration. Mech. Anal. (2001) 157: 91-163. |
[15] | Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal. Real World Appl. (2010) 11: 2302-2313. |
[16] | Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. J. Differential Equations (2016) 261: 2472-2501. |
[17] | Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion. Trans. Amer. Math. Soc. (2016) 368: 8615-8631. |
[18] | Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay. Nonlinear Anal. (2010) 72: 3150-3158. |
[19] | Traveling waves in diffusive random media. J. Dynam. Differential Equations (2004) 16: 1011-1060. |
[20] | Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence. Nonlinearity (2017) 30: 3466-3491. |
[21] | Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J. Differential Equations (2010) 249: 747-795. |
[22] | Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonnlinearity. Nonlinear Anal. (2011) 74: 814-826. |
[23] | Entire solutions for nonlocal dispersal equations with bistable nonlineartiy. J. Differential Equations (2011) 251: 551-581. |
[24] | Entire solutions of a diffusion and competitive Lotka-Volterra type system with nonlocal delayes. Nonlinearity (2010) 23: 1609-1630. |
[25] | J.-B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208 |
[26] | Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case. J. Differential Equations (2015) 258: 2435-2470. |
[27] | Existence of traveling wave solutions for a nonlocal bistable equation: An abstract approach. Publ. Res. Inst. Math. Sci. (2009) 45: 955-979. |
[28] | Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel. Sci. China Math. (2017) 60: 1791-1804. |
[29] | Entire solutions in nonlocal bistable equations: Asymmetric case. Acta Math. Sin. (Engl. Ser.) (2019) 35: 1771-1794. |