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The Trichotomy of Solutions and the Description of Threshold Solutions for Periodic Parabolic Equations in Cylinders

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Abstract

In this paper, the nonnegative bounded solutions for reaction–advection–diffusion equations of the form \(u_{t}-\varDelta u+\alpha (t,y)u_{x}=f(t,y,u)\) in cylinders are studied, where f is a bistable or multistable nonlinearity which is T-periodic in t. We prove that under certain conditions, there are at most three types of solutions for any one-parameter family of initial data: that spread to 1 for large parameters, vanish to 0 for small parameters, and exhibit exceptional behaviors for intermediate parameters. We usually refer to the last as the threshold solutions. It is worth noting that we also give a sufficient condition for solutions to spread to 1 by proving a kind of stability of a pair of diverging traveling fronts. A natural question is what kinds of properties do the threshold solutions have? Under the additional conditions that \(\alpha (t,y)\equiv 0\) and that f and u(0, x, y) are radially symmetric with respect to y around 0 and radially nonincreasing away from 0, by using super- and sub-solutions, Harnack’s inequality and the method of moving hyperplane, we show that any point in the \(\omega \)-limit set of the threshold solutions is symmetric with respect to x, and exponentially decays to 0 as \(|x|\rightarrow \infty \).

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References

  1. Alikakos, N.D., Bates, P.W., Chen, X.: Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans. Am. Math. Soc. 351, 2777–2805 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Aronson, D.G., Weinberger, H.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Lecture Notes in Math. Partial Differential Equations and Related Topics, vol. 446. Springer, New York, pp. 5–49 (1975)

  3. Aronson, D.G., Weinberger, H.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Bao, X., Wang, Z.-C.: Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Voltrra competition system. J. Differ. Equ. 255, 2402–2435 (2013)

    MATH  Google Scholar 

  5. Berestycki, H., Larrouturou, B., Roquejoffre, J.M.: Stability of travelling fronts in a model for flame propagation. Part I: linear analysis. Arch. Ration. Mech. Anal. 117, 97–117 (1991)

    MATH  Google Scholar 

  6. Berestycki, H., Nirenberg, L.: Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. In: Analysis, et cetera. Academic Press, Boston, MA, pp. 115–164 (1990)

  7. Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 22, 1–37 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. NonLinéaire 9, 497–572 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Berestycki, H.: The influence of advection on the propagation of fronts in reaction–diffusion equations. In: Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, Vol. 569, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 11–48 (2002)

  10. Busca, J., Jendoubi, M.A., Poláčik, P.: Convergence to equilibrium for semillinear paraboic problems in \({\mathbb{R}}^{N}\). Commun. Part. Differ. Equ. 27, 1793–1814 (2002)

    MATH  Google Scholar 

  11. Chen, X.-Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blowup in one-dimensional semilinear heat equations. J. Differ. Equ. 78, 160–190 (1989)

    MATH  Google Scholar 

  12. Constantin, P., Kiselev, A., Ryzhik, L.: Quenching of flames by fluid advection. Commun. Pure Appl. Math. 54, 1320–1342 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Contri, B.: Pulsating fronts for bistable on average reaction–diffusion equations in a time periodic environment. J. Math. Anal. Appl. 437, 90–132 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Daners, D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Ding, W., Matano, H.: Dynamics of time-periodic reaction–diffusion equations with compact initial support on \({\mathbb{R}}\). J. Math. Pures Appl. 9(131), 326–371 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Ding, W., Matano, H.: Dynamics of time-periodic reaction–diffusion equations with front-like initial data on \({\mathbb{R}}\). SIAM J. Math. Anal. 52, 2411–2462 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Du, Y., Matano, H.: Convergence and sharp thresholds for propagation in nonlinear diffusion problems. J. Eur. Math. Soc. 12, 279–312 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Du, Y., Lin, Z.: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Du, Y., Poláčik, P.: Locally uniform convergence to an equilibrium for nonlinear parabolic equations on \({\mathbb{R}}^{N}\). Indiana Univ. Math. J. 3, 788–824 (2015)

    MATH  Google Scholar 

  20. Du, Y., Lou, B.: Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17, 2673–2724 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Du, Y., Lou, B., Zhou, M.: Nonlinear diffusion problems with free boundaries: convergence, transition speed, and zero number arguments. SIAM J. Math. Anal. 47, 3555–3584 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Fannjiang, A., Kiselev, A., Ryzhik, L.: Quenching of reaction by cellular flows. Geom. Funct. Anal. 16, 40–69 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Feireisl, E., Petzeltová, H.: Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differ. Integral Equ. 10, 181–196 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Fife, P.C., Mcleod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)

    MathSciNet  MATH  Google Scholar 

  25. Fisher, R.A.: The advance of advantageous genes. Ann. Eugenics 7, 355–369 (1937)

    MATH  Google Scholar 

  26. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  27. Giletti, T., Rossi, L.: Pulsating solutions for multidimensional bistable and multistable equations. Math. Ann. 378, 1555–1611 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Hamel, F., Ninomiya, H.: Localized and expanding entire solutions of reaction–diffusion equations. J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-020-09936-2

  29. Hamel, F., Omrani, S.: Existence of multidimensional travelling fronts with a multistable nonlinearity. Adv. Differ. Equ. 5, 557–582 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Henry, D.: Geometric theory of semilinear parabolic equations. In: Lecture Notes in Mathematics, vol. 840. Springer, New York (1981)

  31. Kanel’Ja, I.: Stabilization of the solutions of the equations of combustion theory with finite initial functions. Mat. Sb. (N.S.) 65, 398–413 (1964)

    MathSciNet  Google Scholar 

  32. Kiselev, A., Zlatoš, A.: Quenching of combustion by shear flows. Duke Math. J. 132, 49–72 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: Étude de l’équation de la chaleur de matiière et son application à un paoblème biologique. Bull. Moskov. Gos. Univ. Mat. Mekh. 1, 1–25 (1937)

    Google Scholar 

  34. Li, W.-T., Liu, N.-W., Wang, Z.-C.: Entire solutions in reaction–advection–diffusion equations in cylinders. J. Math. Pures Appl. 9(90), 492–504 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Liu, N.-W., Li, W.-T., Wang, Z.-C.: Entire solutions of reaction-advection-diffusion equations with bistable nonliearity in cylinders. J. Differ. Equ. 246, 4249–4267 (2009)

    MATH  Google Scholar 

  36. Liu, N.-W., Li, W.-T.: Entire solutions in reaction–advection–diffusion equations with bistable nonlinearities in heterogeneous media. Sci. China Math. 53, 1775–1786 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Berlin (1995)

    MATH  Google Scholar 

  38. Matano, H., Poláčik, P.: Dynamics of nonnegative solutions of one-dimensional reaction–diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences. Commun. Part. Differ. Equ. 41, 785–811 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Poláčik, P.: Symmetry properties of positive solutions of parabolic equations on \({\mathbb{R}}^{N}\). I. Asymptotic symmetry for the Cauchy problem. Commun. Part. Differ. Equ. 30, 1567–1593 (2005)

    MATH  Google Scholar 

  40. Poláčik, P.: Threshold solutions and sharp transitions for nonautonomous parabolic equations on \({\mathbb{R}}^{N}\). Arch. Ration. Mech. Anal. 199, 69–97 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Roquejoffre, J.M.: Stability of travelling fronts in a model for flame propagation. Part II: nonlinear stability. Arch. Ration. Mech. Anal. 117, 119–153 (1992)

    MathSciNet  MATH  Google Scholar 

  42. Roquejoffre, J.M.: Convergence to travelling waves for solutions of a class of semilinear parabolic equations. J. Differ. Equ. 108, 262–295 (1994)

    MathSciNet  MATH  Google Scholar 

  43. Roquejoffre, J.M.: Eventual monotonicity and convergence to travelling fronts for the solution of parabolic equations in cylinders. Ann. Inst. H. Poincaré. Anal. NonLinéaire 14, 499–552 (1997)

    MathSciNet  MATH  Google Scholar 

  44. Vega, J.M.: Travelling waves fronts of reaction–diffusion equations in cylindrical domains. Commun. Part. Differ. Equ. 18, 505–531 (1993)

    MATH  Google Scholar 

  45. Xin, J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)

    MathSciNet  MATH  Google Scholar 

  46. Zel’dovich, Y.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions. Cons. Bureau, New York (1985)

    Google Scholar 

  47. Zhao, G.: Multidimensional periodic traveling waves in infinite cylinders. Discrete Contin. Dyn. Syst. 24, 1025–1045 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Zlatoš, A.: Quenching and propagation of combustion without ignition temperature cutoff. Nonlinearity 18, 1463–1475 (2005)

    MathSciNet  MATH  Google Scholar 

  49. Zlatoš, A.: Sharp transition between extinction and propagation of reaction. J. Am. Math. Soc. 19, 251–263 (2006)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to the anonymous referees for their very valuable comments and suggestions helping to the improvement of the original manuscript. This work was supported by National Natural Science Foundation of China (12071193 and 11731005), Natural Science Foundation of Gansu Province of China (21JR7RA535, 21JR7RA537), and the Heilongjiang Provincial Natural Science Foundation of China (LH2020A003).

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  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Zhuo Ma & Zhi-Cheng Wang

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  1. Zhuo Ma
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Ma, Z., Wang, ZC. The Trichotomy of Solutions and the Description of Threshold Solutions for Periodic Parabolic Equations in Cylinders. J Dyn Diff Equat 35, 3665–3689 (2023). https://doi.org/10.1007/s10884-021-10124-z

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  • DOI: https://doi.org/10.1007/s10884-021-10124-z

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