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Journal of Evolution Equations

, Volume 14, Issue 4–5, pp 749–777 | Cite as

Asymptotics for a nonlinear integral equation with a generalized heat kernel

  • Kazuhiro Ishige
  • Tatsuki KawakamiEmail author
  • Kanako Kobayashi
Article

Abstract

This paper is concerned with a nonlinear integral equation
$$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$
where N ≥  1, \({\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}\) for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as \({t \to \infty}\) .

Keywords

Asymptotic Expansion Heat Kernel Decay Estimate Nonlinear Integral Equation Nonlinear Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Amann H., Fila M.: A Fujita-type theorem for the Laplace equation with a dynamical boundary condition. Acta Math. Univ. Comenianae 66, 321–328 (1997)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Biler P., Funaki T., Woyczynski W.A.: Fractal Burgers equations J. Differential Equations 148, 9–46 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Biler P., Woyczynski W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. 59, 845–869 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Caristi G., MItidieri E.: Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data. J. Math. Anal. Appl. 279, 710–722 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Carpio A.: Large time behaviour in convection-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 551–574 (1996)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cui S.: Local and global existence of solutions to semilinear parabolic initial value problems. Nonlinear Anal. 43, 293–323 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dolbeault J., Karch G.: Large time behavior of solutions to nonhomogeneous diffusion equations. Banach Center Publ. 74, 113–147 (2006)MathSciNetGoogle Scholar
  8. 8.
    Duro G., Zuazua E.: Large time behavior for convection-diffusion equations in R N with asymptotically constant diffusion. Comm. Partial Differential Equations 24, 1283–1340 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Math. Acad. Sci. Paris 330 (2000), 93–98.Google Scholar
  10. 10.
    Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S I. Pohozaev, On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range, C. R. Math. Acad. Sci. Paris 335 (2002), 805–810.Google Scholar
  11. 11.
    Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Differential Equations 9 (2004), 1009–1038.Google Scholar
  12. 12.
    M. FILA, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal. 11 (2012), 1285–1301.Google Scholar
  13. 13.
    Fino A., Karch G.: Decay of mass for nonlinear equation with fractional Laplacian. Monatsh. Math. 160, 375–384 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fujigaki Y., Miyalawa T.: Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space. SIAM J. Math. Anal. 33, 523–544 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J. 51 (2002), 1321–1338.Google Scholar
  16. 16.
    F. Gazzola and H.-C. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var. Partial Differential Equations 30 (2007), 389–415.Google Scholar
  17. 17.
    A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in R N, J. Differential Equations 53 (1984), 258–276.Google Scholar
  18. 18.
    Hayashi N., Kaikina E.I., Naumkin P.I.: Asymptotics for fractional nonlinear heat equations. J. London Math. Soc. 72, 663–688 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Herraiz L.A.: Asymptotic behaviour of solutions of some semilinear parabolic problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 49–105 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    K. Ishige, M. ISHIWATA and T. Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J. 58 (2009), 2673–2708.Google Scholar
  21. 21.
    Ishige K., Kawakami T.: Asymptotic behavior of solutions for some semilinear heat equations in R N. Commun. Pure Appl. Anal. 8, 1351–1371 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ishige K., Kawakami T.: Refined asymptotic profiles for a semilinear heat equation. Math. Ann. 353, 161–192 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math. 121 (2013), 317–351.Google Scholar
  24. 24.
    K. Ishige, T. Kawakami and K. Kobayashi, Global solutions for a nonlinear integral equation with a generalized heat kernel, Discrete Contin. Dyn. Syst. Ser. S. 7 (2014), 767–783.Google Scholar
  25. 25.
    K. Ishige and K. Kobayashi, Convection-diffusion equation with absorption and non-decaying initial data, J. Differential Equations 254 (2013), 1247–1268.Google Scholar
  26. 26.
    S Kamin., L. A. Peletier , Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 393–408.Google Scholar
  27. 27.
    T. Ogawa and M. Yamamoto, Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation, Math. Models Methods Appl. Sci. 19 (2009), 939–967.Google Scholar
  28. 28.
    P. Quittner and P. Souplet, Superlinear parabolic problems: Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Basel, 2007.Google Scholar
  29. 29.
    Raczyński A.: Diffusion-dominated asymptotics of solution to chemotaxis model. J. Evol. Equ. 11, 509–529 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Sugitani S.: On nonexistence of global solutions for some nonlinear integral equations. Osaka J. Math. 12, 45–51 (1975)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Taskinen J.: Asymptotical behaviour of a class of semilinear diffusion equations. J. Evol. Equ. 7, 429–447 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Yamada T.: Higher-order asymptotic expansions for a parabolic system modeling chemotaxis in the whole space. Hiroshima Math. J. 39, 363–420 (2009)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Yamada T.: Moment estimates and higher-order asymptotic expansions of solutions to a parabolic system in the whole space. Funkcial. Ekvac. 54, 15–51 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Yamamoto M.: Asymptotic expansion of solutions to the drift-diffusion equation with large initial data. J. Math. Anal. Appl. 369, 144–163 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Yamamoto M.: Asymptotic expansion of solutions to the dissipative equation with fractional Laplacian. SIAM J. Math. Anal. 44, 3786–3805 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Kazuhiro Ishige
    • 1
  • Tatsuki Kawakami
    • 2
    Email author
  • Kanako Kobayashi
    • 1
  1. 1.Mathematical InstituteTohoku UniversityAobaJapan
  2. 2.Department of Mathematical SciencesOsaka Prefecture UniversitySakaiJapan

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