## Taiwanese Journal of Mathematics

### Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations

#### Abstract

The first aim in the paper is to prove the local exponential asymptotic stability of some entire solutions to bistable reaction diffusion equations via the super-sub solution method. If the integral of the reaction term $f$ over the interval $[0,1]$ is positive, we not only obtain the similar asymptotic stability result found by Yagisita in 2003, but also simplify the proof. The asymptotic stability result for the case $\int^1_{0} f(u) \, du \lt 0$ is also obtained, which is not considered by Yagisita. After that, the asymptotic behavior of entire solutions as $t \to +\infty$ is investigated, since the other side was completely known. Here, the result is established by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions, instead of constructing the super-sub solutions as usual. Finally, for the special bistable case $f(u) = u(1-u)(u-\alpha)$, $\alpha \in (0,1)$, we prove the entire solution continuously depends on $\alpha$.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 22 pages.

Dates
First available in Project Euclid: 11 May 2020

https://projecteuclid.org/euclid.twjm/1589184077

Digital Object Identifier
doi:10.11650/tjm/200502

#### Citation

Wang, Yang; Li, Xiong. Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations. Taiwanese J. Math., advance publication, 11 May 2020. doi:10.11650/tjm/200502. https://projecteuclid.org/euclid.twjm/1589184077

#### References

• D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5–49, Lecture Notes in Mathematics 446, Springer, Berlin, 1975.
• ––––, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76.
• X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212 (2005), no. 1, 62–84.
• Y.-Y. Chen, J.-S. Guo, H. Ninomiya and C.-H. Yao, Entire solutions originating from monotone fronts to the Allen-Cahn equation, Phys. D 378/379 (2018), 1–19.
• H. Cohen, Nonlinear diffusion problems, Studies in Applied Mathematics, 27–64, MAA Studies in Mathematics 7, Math. Assoc. of Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1971.
• P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin, 1979.
• ––––, Long time behavior of solutions of bistable nonlinear diffusion equations, Arch. Rational Mech. Anal. 70 (1979), no. 1, 31–46.
• P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361.
• ––––, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1981), no. 4, 281–314.
• R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), no. 4, 355–369.
• Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Proceedings of Third East Asia Partial Differential Equation Conference, Taiwanese J. Math. 8 (2004), no. 1, 15–32.
• T. Gallay and E. Risler, A variational proof of global stability for bistable travelling waves, Differential Integral Equations 20 (2007), no. 8, 901–926.
• J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 193–212.
• F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999), no. 10, 1255–1276.
• ––––, Travelling fronts and entire solutions of the Fisher-KPP equation in R$^N$, Arch. Ration. Mech. Anal. 157 (2001), no. 2, 91–163.
• Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math. 13 (1996), no. 1, 117–133.
• A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Étude de \'Iequation de la diffusion avec croissance de la quantité de matière et son application à unprobleme biologique, Bull. Univ. Moskou, Ser. Internat., Sec. A 1 (1937), no. 6, 1–25.
• C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation II, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1901–1924.
• H. Matano and P. Poláčik, An entire solution of a bistable parabolic equation on $\mathbb{R}$ with two colliding pulses, J. Funct. Anal. 272 (2017), no. 5, 1956–1979.
• Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations 18 (2006), no. 4, 841–861.
• J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines, IEEE Trans. Circuit Theory 12 (1965), no. 3, 400–412.
• T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 1–34.
• ––––, Stability analysis in order-preserving systems in the presence of symmetry, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 2, 395–438.
• E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 2, 381–424.
• V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 3, 341–379.
• Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2047–2084.
• J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Comm. Partial Differential Equations 17 (1992), no. 11-12, 1889–1899.
• H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci. 39 (2003), no. 1, 117–164.