Tohoku Mathematical Journal

Homoclinic and heteroclinic orbits for a semilinear parabolic equation

Marek Fila and Eiji Yanagida

Full-text: Open access

Abstract

We study the existence of connecting orbits for the Fujita equation with a critical or supercritical exponent. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and a homoclinic orbit with respect to zero.

Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 561-579.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886281

Digital Object Identifier
doi:10.2748/tmj/1325886281

Mathematical Reviews number (MathSciNet)
MR2872956

Zentralblatt MATH identifier
1252.35157

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 35V05

Keywords
Homoclinic orbit heteroclinic orbits ancient solutions stationary solutions self-similar solutions

Citation

Fila, Marek; Yanagida, Eiji. Homoclinic and heteroclinic orbits for a semilinear parabolic equation. Tohoku Math. J. (2) 63 (2011), no. 4, 561--579. doi:10.2748/tmj/1325886281. https://projecteuclid.org/euclid.tmj/1325886281


Export citation

References

  • T. Bartsch, P. Poláčik and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations, J. Eur. Math. Soc. 13 (2011), 219–247.
  • M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions, 189–198, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998.
  • P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynam. Report. Ser. Dynam. Systems Appl. 1 (1988), 57–89.
  • P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution, J. Differential Equations 81 (1989), 106–135.
  • N. Chafee and E. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37.
  • B. Fiedler, Global attractors of one-dimensional parabolic equations: sixteen examples, Tatra Mt. Math. Publ. 4 (1994), 67–92.
  • B. Fiedler and C. Rocha, Heteroclinic orbits of scalar semilinear parabolic equations, J. Differential Equations 125 (1996), 239–281.
  • B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Differential Equations 156 (1999), 282–308.
  • B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc. 352 (2000), 257–284.
  • M. Fila, J. R. King, M. Winkler and E. Yanagida, Linear behaviour of solutions of a superlinear heat equation, J. Math. Anal. Appl. 340 (2008), 401–409.
  • M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Discrete Contin. Dynam. Systems 6 (2000), 155–164.
  • M. Fila, H. Matano and P. Poláčik, Existence of $L^1$-connections between equilibria of a semilinear parabolic equation, J. Dynam. Differential Equations 14 (2002), 463–491.
  • M. Fila and N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations 20 (2007), 671–680.
  • M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations 4 (1999), 163–196.
  • M. Fila, M. Winkler and E. Yanagida, Convergence to selfsimilar solutions for a semilinear parabolic equation, Discrete Contin. Dyn. Syst. 21 (2008), 703–716.
  • A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425–447.
  • Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math. 8 (2004), 15–32.
  • G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Differential Equations 91 (1991), 111–137.
  • V. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1–67.
  • B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525–598.
  • A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167–189.
  • D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, New York, 1981.
  • D. Henry, Some infinite dimensional Morse–Smale systems defined by parabolic differential equations, J. Differential Equations 59 (1985), 165–205.
  • L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial'nye Uravneniya 24 (1988), 1226–1234 (English translation: Differential Equations 24 (1988), 799–805).
  • L. A. Lepin, Self-similar solutions of a semilinear heat equation, (in Russian), Mat. Model. 2 (1990), 63–74.
  • Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p=0$ in $\boldsymbol{R}^n$, J. Differential Equations 95 (1992), 304–330.
  • N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal. 257 (2009), 2911–2937.
  • Y. Naito, An ODE approach to the multiplicity of self-similar solutions for semilinear heat equations, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 807–835.
  • P. Poláčik, Parabolic equations: Asymptotic behavior and dynamics on invariant manifolds, Handbook of dynamical systems (B. Fiedler ed.), Vol. 2, Chapter 16, Elsevier, Amsterdam, 2002.
  • P. Poláčik and P. Quittner, A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation, Nonlinear Anal. 64 (2006), 1679–1689.
  • P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations, Indiana Univ. Math. J. 56 (2007), 879–908.
  • P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745–771.
  • P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations 208 (2005), 194–214.
  • P. Quittner and Ph. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.
  • J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren Math. Wiss. 258, Springer-Verlag, New York, 1983.
  • Ph. Souplet and F. B. Weissler, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 213–235.
  • X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), 549–590.