# zbMATH — the first resource for mathematics

Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations. (English) Zbl 1364.35145
Summary: We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $$\mathbb{R}^N$$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $$L^2$$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

##### MSC:
 35K57 Reaction-diffusion equations 35K15 Initial value problems for second-order parabolic equations 35A15 Variational methods applied to PDEs 35B07 Axially symmetric solutions to PDEs
##### Keywords:
sharp transition; traveling waves; gradient flow
Full Text:
##### References:
 [1] S. M. Allen, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta Metal., 27, 1085, (1979) [2] D. G. Aronson, Multidimensional diffusion arising in population genetics,, Adv. Math., 30, 33, (1978) · Zbl 0407.92014 [3] R. Bamón, Ground states of semilinear elliptic equations: A geometric approach,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, 551, (2000) · Zbl 0988.35054 [4] P. W. Bates, Existence and instability of spike layer solutions to singular perturbation problems,, J. Funct. Anal., 196, 211, (2002) · Zbl 1010.47036 [5] H. Berestycki, Nonlinear scalar field equations. I. Existence · Zbl 0533.35029 [6] H. Berestycki, An ODE approach to the existence of positive solutions for semilinear problems in $$\mathbb R^n$$,, Indiana Univ. Math. J., 30, 141, (1981) · Zbl 0522.35036 [7] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $$R^n$$ or $$R^n_+$$ through the method of moving planes,, Comm. Partial Differential Equations, 22, 1671, (1997) · Zbl 0910.35048 [8] J. Busca, Convergence to equilibrium for semilinear parabolic problems in $$\mathbbR^N$$,, Comm. Partial Differential Equations, 27, 1793, (2002) · Zbl 1021.35013 [9] X. Cabré, On the stability of radial solutions of semilinear elliptic equations in all of $$\mathbbR^n$$,, C. R. Math. Acad. Sci. Paris, 338, 769, (2004) · Zbl 1081.35029 [10] X. Cabré, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58, 1678, (2005) · Zbl 1102.35034 [11] L. A. Caffarelli, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42, 271, (1989) · Zbl 0702.35085 [12] A. Capella-Kort, Stable Solutions of Nonlinear Elliptic Equations: Qualitative and Regularity Properties,, PhD thesis, (2005) [13] E. N. Dancer, Some remarks on Liouville type results for quasilinear elliptic equations,, Proc. Amer. Math. Soc., 131, 1891, (2003) · Zbl 1076.35038 [14] Y. Du, Convergence and sharp thresholds for propagation in nonlinear diffusion problems,, J. Eur. Math. Soc., 12, 279, (2010) · Zbl 1207.35061 [15] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998) · Zbl 0902.35002 [16] E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on $$\mathbb R$$,, Comment. Math. Univ. Carolinae, 39, 525, (1998) · Zbl 0963.35080 [17] E. Feireisl, On the long time behaviour of solutions to nonlinear diffusion equations on $$R^n$$,, Nonlin. Diff. Eq. Appl., 4, 43, (1997) · Zbl 0872.35014 [18] E. Feireisl, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations,, Differential Integral Equations, 10, 181, (1997) · Zbl 0879.35023 [19] P. C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equations,, Arch. Rational Mech. Anal., 70, 31, (1979) · Zbl 0435.35045 [20] J. Földes, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $$\mathbbR^N$$,, J. Differential Equations, 251, 1903, (2011) · Zbl 1263.35035 [21] A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964) · Zbl 0144.34903 [22] V. A. Galaktionov, On convergence in gradient systems with a degenerate equilibrium position,, Mat. Sb., 198, 65, (2007) · Zbl 1229.35078 [23] D. Gilbarg, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983) · Zbl 0361.35003 [24] C. Gui, On the stability and instability of positive steady states of a semilinear heat equation in $$R^n$$,, Comm. Pure Appl. Math., 45, 1153, (1992) · Zbl 0811.35048 [25] C. K. R. T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions,, Rocky Mountain J. Math., 13, 355, (1983) · Zbl 0528.35054 [26] C. K. R. T. Jones, Spherically symmetric solutions of a reaction-diffusion equation,, J. Diff. Equations, 49, 142, (1983) · Zbl 0523.35059 [27] Y. I. Kanel’, On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combusion,, Mat. Sbornik, 59, 245, (1962) [28] B. S. Kerner, Autosolitons,, Kluwer, (1994) [29] E. H. Lieb, Analysis,, American Mathematical Society, (1997) [30] C. S. Lin, A counterexample to the nodal domain conjecture and a related semilinear equation,, Proc. Amer. Math. Soc., 102, 271, (1988) · Zbl 0652.35085 [31] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress in Nonlinear Differential Equations and their Applications,, Birkhäuser, (1995) · Zbl 0816.35001 [32] H. P. McKean, Nagumo’s equation,, Adv. Math., 4, 209, (1970) · Zbl 0202.16203 [33] A. G. Merzhanov, Physics of reaction waves,, Rev. Mod. Phys., 71, 1173, (1999) [34] A. S. Mikhailov, Foundations of Synergetics,, Springer-Verlag, (1990) · Zbl 0712.92001 [35] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type,, Discrete Contin. Dyn. Syst. Ser. B, 4, 867, (2004) · Zbl 1069.35031 [36] C. B. Muratov, Front propagation in infinite cylinders. I. A variational approach,, Comm. Math. Sci., 6, 799, (2008) · Zbl 1173.35537 [37] C. B. Muratov, Global stability and exponential convergence to variational traveling waves in cylinders,, SIAM J. Math. Anal., 44, 293, (2012) · Zbl 1254.35213 [38] C. B. Muratov, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations,, Nonlin. Diff. Eq. Appl., 20, 1519, (2013) · Zbl 1433.35173 [39] J. D. Murray, Mathematical Biology,, Springer-Verlag, (1989) · Zbl 0682.92001 [40] J. Nagumo, An active pulse transmission line simulating nerve axon,, Proc. IEEE, 50, 2061, (1962) [41] P. Poláčik, Morse indices and bifurcations of positive solutions of $$Δ u+f(u)=0$$ on $$\mathbbR^N$$,, Indiana Univ. Math. J., 50, 1407, (2001) · Zbl 1101.35321 [42] P. Poláčik, Nonconvergent bounded trajectories in semilinear heat equations,, J. Differential Equations, 124, 472, (1996) · Zbl 0845.35054 [43] P. Poláčik, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics,, SIAM J. Math. Anal., 46, 3481, (2014) · Zbl 1316.35148 [44] P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $$\mathbbR^n$$,, Arch. Ration. Mech. Anal., 199, 69, (2011) · Zbl 1262.35130 [45] P. Quittner, Superlinear Parabolic Problems,, Birkhäuser Advanced Texts: Basler Lehrbücher., (2007) [46] V. Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 341, (2004) · Zbl 1066.35018 [47] J. Serrin, Uniqueness of ground states for quasilinear elliptic equations,, Indiana Univ. Math. J., 49, 897, (2000) · Zbl 0979.35049 [48] J. Shi, Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations,, J. Differential Equations, 231, 235, (2006) · Zbl 1387.35333 [49] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems,, Annals Math., 118, 525, (1983) · Zbl 0549.35071 [50] M. Tang, Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations,, J. Differential Equations, 164, 155, (2000) · Zbl 0961.34007 [51] K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients,, Arch. Rational Mech. Anal., 90, 291, (1985) · Zbl 0618.35058 [52] J. Xin, Front propagation in heterogeneous media,, SIAM Review, 42, 161, (2000) · Zbl 0951.35060 [53] A. Zlatoš, Sharp transition between extinction and propagation of reaction,, J. Amer. Math. Soc., 19, 251, (2006) · Zbl 1081.35011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.