Lifespan, asymptotic behavior and ground-state solutions to a nonlocal parabolic equation

  • Jun ZhouEmail author


We consider a nonlocal parabolic equation, which was studied in Liu and Ma (Nonlinear Anal 110:141–156, 2014), Li and Liu (J Math Phys 58:101503, 2017). By exploiting the boundary condition and the variational structure of the equation, we obtain the lifespan for the blow-up solutions, decay estimation and asymptotic behavior for the global solutions and ground-state solutions for the stationary solutions. The results generalize the former studies on this equation.


Nonlocal parabolic equation Lifespan Decay estimation Ground-state solution 

Mathematics Subject Classification

35K20 35K35 35B40 



  1. 1.
    Bao, A.G., Song, X.F.: Bounds for the blowup time of the solutions to quasi-linear parabolic problems. Z. Angew. Math. Phys. 65, 115–123 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Badiale, M., Serra, E.: Semilinear elliptic equations for beginners-existence results via the variational approach. In: Axler, S., Capasso, V., Casacuberta, C., Macintyre, A.J., Ribet, K., Sabbah, C., Süli, E., Woyczynski, W. (eds.) Universitext. Springer, London (2011) Google Scholar
  3. 3.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Gourley, S.A.: Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41(3), 272–284 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic beating: part 1. Model derivation and some special cases. Eur. J. Appl. 6(2), 127–144 (1995)CrossRefGoogle Scholar
  6. 6.
    Levine, H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form \(Pu_{tt} = Au + F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)zbMATHGoogle Scholar
  7. 7.
    Liu, B.Y., Ma, L.: Invariant sets and the blow up threshold for a nonlocal equation of parabolic type. Nonlinear Anal. 110, 141–156 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (1977)Google Scholar
  9. 9.
    Li, X.L., Liu, B.Y.: Vacuum isolating, blow up threshold and asymptotic behavior of solutions for a nonlocal parabolic equation. J. Math. Phys. 58, 101503 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu, Y., Luo, S., Ye, Y.: Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions. Comput. Math. Appl. 65, 1194–1199 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Liu, Y.: Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition. Comput. Math. Appl. 66, 2092–2095 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ni, W.M., Sacks, P.E., Tavantzis, J.: On the asymptotic behavior of solutions of certain quasilinear parabolic equations. J. Differ. Equ. 54, 97–120 (1984)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ou, C., Wu, J.: Persistence of wavefronts in delayed nonlocal reaction–diffusion equations. J. Differ. Equ. 235(1), 219–261 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Payne, L.E., Schaefer, P.W.: Bounds for blow-up time for the heat equation under nonlinear boundary conditions. Proc. R. Soc. Edinb. 139A, 1289–1296 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Payne, L.E., Philippin, G.A.: Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions. Proc. Am. Math. Soc. 141(7), 2309–2318 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Quittner, P., Souplet, P.: Superlinear Parabolic Problems: Blow-Up, Gobal Existence and Steady States. Birkhäuser Advanced Texts, Basler Lehrbücher. Springer, Basel (2007)zbMATHGoogle Scholar
  17. 17.
    So, J.W.H., Wu, J.H., Zou, X.F.: A reaction–diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains. Proc. Lond. Math. Soc. A 457, 1841–1853 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Song, J.C.: Lower bounds for the blow-up time in a non-local reaction–diffusion problem. Appl. Math. Lett. 24, 793–796 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Xu, G.Y., Zhou, J.: Lifespan for a semilinear pseudo-parabolic equation. Math. Methods Appl. Sci. 41(2), 705–713 (2018)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhou, J.: Lower bounds for blow-up time of two nonlinear wave equations. Appl. Math. Lett. 45, 64–68 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhou, J.: Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level. Appl. Math. Lett. 78, 118–125 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China

Personalised recommendations