The stability of the equilibria of the Allen–Cahn equation with fractional diffusion

References

• Aronson DG , Weinberger HF . Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial differential equations and related topics. Vol. 466, Lecture notes in mathematics. Berlin: Springer; 1975. p. 5–49.
   Google Scholar
• Aronson DG , Weinberger HF . Multidimensional nonlinear diffusion arising in population genetics. Adv Math. 1978;30:33–76.
   Web of Science ®Google Scholar
• Volpert AI , Volpert VA , Volpert VA . Traveling wave solutions of parabolic systems. Providence (RI): American Mathematical Society; 1994.
   Google Scholar
• Ninomiya H , Taniguchi M . Existence and global stability of traveling curved fronts in the Allen--Cahn equations. J Differ Equ. 2005;213:204–233.
   Web of Science ®Google Scholar
• Ninomiya H , Taniguchi M . Global stability of traveling curved fronts in the Allen--Cahn equations. Discrete Contin Dyn Syst. 2006;15:819–832.
   Web of Science ®Google Scholar
• Taniguchi M . Traveling fronts of pyramidal shapes in the Allen--Cahn equations. SIAM J Appl Math. 2007;39:319–344.
   Web of Science ®Google Scholar
• Taniguchi M . The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen--Cahn equations. J Differ Equ. 2009;246:2103–2130.
   Web of Science ®Google Scholar
• Landkof NS . Foundations of modern potential theory. In: Doohovskoy AP , editor. Die Grundlehren der mathematischen Wissenschaften, Band 180. New York (NY): Springer; 1972. Translated from the Russian.
   Google Scholar
• Berestycki H , Roquejoffre J-M , Rossi L . The periodic patch model for population dynamics with fractional diffusion. Discrete Contin Dyn Syst Ser S. 2011;4:1–13.
   Web of Science ®Google Scholar
• Cabré X , Sire Y . Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann Inst H Poincaré Anal Non Linéaire. 2014;31:23–53.
   Web of Science ®Google Scholar
• Cabré X , Sire Y . Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans Amer Math Soc. 2015;367:911–941.
   Web of Science ®Google Scholar
• Caffarelli L , Silvestre L . An extension problem related to the fractional Laplacian. Commun Partial Differ Equ. 2007;32:1245–1260.
   Web of Science ®Google Scholar
• Esen A , Yagmurlu NM , Tasbozan O . Approximate analytical solution to time-fractional damped burger and Cahn--Allen equations. Appl Math Inf Sci. 2013;7:1951–1956.
   Google Scholar
• Fazly M , Sire Y . Symmetry results for fractional elliptic systems and related problems. Commun Partial Differ Equ. 2015;40:1070–1095.
   Web of Science ®Google Scholar
• Mainardi F , Luchko Y , Pagnini G . The fundamental solution of the space-time fractional diffusion equation. Frac Calc Appl Anal. 2001;4:153–192.
   Google Scholar
• Zaslavsky GM . Chaos, fractional kinetics, and anomalous transport. Phys Rep. 2002;371:461–580.
   Web of Science ®Google Scholar
• Cabré X , Roquejoffre J-M . The influence of fractional diffusion in Fisher-KPP equations. Commun Math Phys. 2013;320:679–722.
   Web of Science ®Google Scholar
• Engler H . On the speed of spread for fractional reaction--diffusion equations. Int J Differ Equ. 2010;2009. ID 315421 (16 pages).
   Google Scholar
• Nec Y , Nepomnyashchy A , Golovin A . Front-type solutions of fractional Allen--Cahn equation. Phys D Nonlinear Phenom. 2008;237:3237–3251.
   Web of Science ®Google Scholar
• Gui C , Huan T . Traveling wave solutions to some reaction diffusion equations with fractional Laplacians. Calc Var Partial Differ Equ. 2015;54:251–273.
   Web of Science ®Google Scholar
• Gui C , Zhao M . Traveling wave solutions of Allen--Cahn equation with a fractional Laplacian. Ann Inst H Poincaré Anal Non Linéaire. 2015;32:785–812.
   Web of Science ®Google Scholar
• Pazy A . Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer Science & Business Media; 1983.
   Google Scholar
• Kolokoltsov V . Symmetric stable laws and stable-like jump-diffusions. Proc London Math Soc. 2000;80:725–768.
   Web of Science ®Google Scholar