zbMATH — the first resource for mathematics

Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on \({\mathbb{R}}\). (English) Zbl 1319.35097
The author considers bounded solutions of the Cauchy problem for the semilinear parabolic equation \(u_t=u_{xx}+f(u)\) on \(\mathbb R\). It is showed that there always exist bounded solutions whose \(\omega\)-limit set with respect to the locally uniform convergence contains functions which are not steady states. For balanced bistable nonlinearities, there are examples of such solutions with initial values \(u(x,0)\) converging to \(0\) as \(|x|\) goes to infinity. A considered example with an unbalanced bistable nonlinearity shows that bounded solutions whose \(\omega\)-limit set does not consist of steady states occur for a robust class of nonlinearities \(f\).

35K58 Semilinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Angenent, S., The zeroset of a solution of a parabolic equation, J. reine angew, Math., 390, 79-96, (1988) · Zbl 0644.35050
[2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-345. · Zbl 0533.35029
[3] J. Busca, M.-A. Jendoubi, and P. Poláčik, Convergence to equilibrium for semilinear parabolic problems in\({\mathbb{R}^N}\), Comm. Partial Differential Equations 27 (2002), 1793-1814. · Zbl 1021.35013
[4] X. Chen and J.-S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212 (2005), 62-84. · Zbl 1079.35046
[5] X. Chen, J.-S. Guo, and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 1207-1237. · Zbl 1123.35024
[6] Chen, X.-Y., A strong unique continuation theorem for parabolic equations, math, Ann., 311, 603-630, (1998) · Zbl 0990.35028
[7] X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), 160-190. · Zbl 0692.35013
[8] P. Collet and J.-P. Eckmann, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity 5 (1992), 1265-1302. · Zbl 0757.35059
[9] C. Cortázar, M. del Pino, and M. Elgueta, The problem of uniqueness of the limit in a semilinear heat equation, Comm. Partial Differential Equations 24 (1999), 2147-2172. · Zbl 0940.35107
[10] Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. European Math. Soc. 12 (2010), no. 2, 279-312. · Zbl 1207.35061
[11] Y. Du and P. Poláčik, Locally uniform convergence to an equilibrium for nonlinear parabolic equations on\({\mathbb{R}^N}\), Indiana Univ. Math. J. (to appear).
[12] J.-P. Eckmann and J. Rougemont, Coarsening by Ginzburg-Landau dynamics, Comm. Math. Phys. 199 (1998), no. 2, 441-470. · Zbl 1057.35508
[13] E. Fašangová, Asymptotic analysis for a nonlinear parabolic equation on\({{\mathbb{R}}}\), Comment. Math. Univ. Carolinae 39 (1998), 525-544.
[14] E. Feireisl, On the long time behavior of solutions to nonlinear diffusion equations on\({{\mathbb{R}}^{N}}\), NoDEA Nonlinear Differential Equations Appl. 4 (1997), 43-60. · Zbl 1174.35403
[15] E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations 10 (1997), 181-196. · Zbl 0879.35023
[16] E. Feireisl and P. Poláčik, Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on\({\mathbb{R}}\), Adv. Differential Equations 5 (2000), 583-622. · Zbl 0987.35079
[17] B. Fiedler and P. Brunovský, Connections in scalar reaction diffusion equations with Neumann boundary conditions, Equadiff 6 (Brno, 1985), Lecture Notes in Math., vol. 1192, Springer, Berlin, 1986, pp. 123-128.
[18] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations 125 (1996), 239-281. · Zbl 0849.35056
[19] 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) · Zbl 0361.35035
[20] P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1980/81), no. 4, 281-314. · Zbl 0459.35044
[21] J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on\(R\)\^{}{\(N\)}, J. Differential Equations 251 (2011), 1903-1922. · Zbl 1263.35035
[22] T. Gallay and S. Slijepčević, Energy flow in extended gradient partial differential equations, J. Dynam. Differential Equations 13 (2001), 757-789. · Zbl 1003.35085
[23] T. Gallay and S. Slijepčević, Distribution of energy and convergence to equilibria in extended dissipative systems, J. Dynam. Differential Equations, to appear. · Zbl 1338.35045
[24] J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dynam. Systems 12 (2005), 193-212. · Zbl 1065.35130
[25] F. Hamel, R. Monneau, and J-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4) 37 (2004), 469-506. · Zbl 1085.35075
[26] F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999), no. 10, 1255-1276. · Zbl 0932.35113
[27] F. Hamel and Y. Sire, Spreading speeds for some reaction-diffusion equations with general initial conditions, SIAM J. Math. Anal. 42 (2010), 2872-2911. · Zbl 1259.35054
[28] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Urall’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968, Russian Original: “Nauka”, Moscow 1967. · Zbl 1138.35042
[29] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221-227. · Zbl 0387.35008
[30] H. Matano and P. Poláčik, Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data, in preparation. · Zbl 1345.35052
[31] Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations 18 (2006), 841-861. · Zbl 1125.35050
[32] Y. Morita and H. Ninomiya, Traveling wave solutions and entire solutions to reaction-diffusion equations, Sugaku Expositions 23 (2010), 213-233. · Zbl 1225.35047
[33] P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on\({\mathbb{R}^N}\), Arch. Rational Mech. Anal. 199 (2011), 69-97, Addendum: www.math.umn.edu/ polacik/Publications. · Zbl 1262.35130
[34] P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745-771. · Zbl 1055.35055
[35] Poláčik, P.; Yanagida, E., Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation, Differential Integral Equations, 17, 535-548, (2004) · Zbl 1174.35403
[36] P. Poláčik and E. Yanagida, Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics, SIAM J. Math. Anal, to appear. · Zbl 1316.35148
[37] Roquejoffre, J.-M.; Roussier-Michon, V., Nontrivial large-time behaviour in bistable reaction-diffusion equations, ann, Mat. Pura Appl., 4, 207-233, (2009) · Zbl 1178.35074
[38] Rougemont, J., Dynamics of kinks in the Ginzburg-Landau equation: approach to a metastable shape and collapse of embedded pairs of kinks, Nonlinearity, 12, 539-554, (1999) · Zbl 0984.35148
[39] Yanagida, E., Irregular behavior of solutions for fisher’s equation, J, Dynam. Differential Equations, 19, 895-914, (2007) · Zbl 1138.35042
[40] Zelenyak T., I., Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential Equations, 4, 17-22, (1968) · Zbl 0232.35053
[41] Zlatoš, A., Sharp transition between extinction and propagation of reaction, J, Amer. Math. Soc., 19, 251-263, (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.