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Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1143–1162 | Cite as

Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

  • Hua ChenEmail author
  • Hui Yang XuEmail author
Article
  • 19 Downloads

Abstract

In this paper, we study the initial-boundary value problem for the semilinear pseudo-parabolic equations ut − ΔXut − ΔXu = |u|p−1u, where X = (X1,X2,…,Xm) is a system of real smooth vector fields which satisfy the Hörmander’s condition, and ΔX = \({{\rm{\Delta}}_X} = \sum\nolimits_{j = 1}^m {X_j^2} \) is a finitely degenerate elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.

Keywords

Finitely degenerate pseudo-parabolic equation global existence blow-up decay estimate 

MR(2010) Subject Classification

35K55 35K65 

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Notes

Acknowledgements

The authors are highly grateful to the anonymous referee for his/her careful reading of the manuscript and valuable comments.

References

  1. [1]
    Ball, J. M.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford Ser., 28, 473–486 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Benedetto, E. D., Pierre, M.: On the maximum principle for pseudoparabolic equations. Indiana Univ. Math. J., 30, 821–854 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bony, J. M.: Principe du maximum, inégalité de Harnack et unicité du probléme de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier, 19, 277–304 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bramanti, M.: An Invitation to Hypoelliptic Operators and Hormander’s Vector Fields, Springer Briefs in Mathematics, 2014CrossRefzbMATHGoogle Scholar
  5. [5]
    Cao, Y., Yin, J. X., Wang, C. P.: Cauchy problems of semilinear pseudo-parabolic equations. J. Differential Equations, 246, 4568–4590 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Capogna, L., Danielli, D., Garofalo, N.: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Comm. Partial Differential Equations, 18, 1765–1794 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen, H., Luo, P.: Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators. Calc. Var. Partial Differ. Equ., 54, 2831–2852 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Chen, H., Tian, S. Y.: Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differential Equations, 285, 4424–4442 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Evans, L. C.: Partial Differential Equations, Amer. Math. Soc., 2015Google Scholar
  10. [10]
    Fefferman, C., Phong, D.: Subelliptic eigenvalue problems, in: Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund, 590–606, Wadsworth Math. Series, 1981Google Scholar
  11. [11]
    Ghezzi, R., Jean, F.: Hausdorff volume in non equiregular sub-Riemannian manifolds. Nonlinear Anal., 126, 345–377 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Gopala Rao, V. R., Ting, T. W.: Solutions of pseudo-heat equations in the whole space. Arch. Ration. Mech. Anal., 49, 57–78 (1972/73)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré, Mem. Amer. Math. Soc., 1998zbMATHGoogle Scholar
  14. [14]
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math., 119, 147–171 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Jerison, D.: The Poincaré inequality for vector fields satisfying Hormander’s condition. Duke Math. J., 53, 503–523 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Jerison, D., Sánchez-Calle, A.: Subelliptic, second order differential operators, in: Complex Analysis III, 46–77, 1987CrossRefGoogle Scholar
  17. [17]
    Kohn, J. J.: Hypoellipticity of some degenerate subelliptic operators. J. Funct. Anal., 159, 203–216 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Kohn, J. J.: Subellipticity of the \(\overline \partial \)-Neumann problem on pseudo-convex domains: sufficient conditions. Acta Math., 142, 79–122 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Komornik, V.: Exact Controllability and Stabilization, The Multiplier Method, Masson, John Wiley, Paris, Chichester, 1994zbMATHGoogle Scholar
  20. [20]
    Korpusov, M. O., Sveshnikov, A. G.: Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics. Zh. Vychisl. Mat. Mat. Fiz., 43, 1835–1869 (2003) (in Russian); transl. in: Comput. Math. Math. Phys., 43, 1765–1797 (2003)MathSciNetzbMATHGoogle Scholar
  21. [21]
    Korpusov, M. O., Sveshnikov, A. G.: Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources. Differ. Uravn., 42, 404–415 (2006) (in Russian); transl. in: Differ. Equ., 42, 431–443 (2006)MathSciNetzbMATHGoogle Scholar
  22. [22]
    Levine, H. A.: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(\mathscr{P}u_{t}=-\mathscr{A}u+\mathscr{F}(u)\). Arch. Ration. Mech. Anal., 51, 371–386 (1973)CrossRefGoogle Scholar
  23. [23]
    Lewy, H.: An example of a smooth linear partial differential equation without solution. Ann. Math., 66, 155–158 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Liu, Y. C., Zhao, J. S.: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal., 64, 2665–2687 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Luo, P.: Blow-up phenomena for a pseudo-parabolic equation. Math. Methods Appl. Sci., 38, 2636–2641 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Métivier, G.: Fonction spectrale et valeurs propres d’une classe d’opérateurs non elliptiques. Comm. Partial Differential Equations, 1, 467–519 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications, Amer. Math. Soc., 2002zbMATHGoogle Scholar
  28. [28]
    Morbidelli, D.: Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields. Studia Mathematica, 139, 213–244 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Nagel, A., Stein, E. M., Wainger, S.: Balls and metrics defined by vector fields I: Basic properties. Acta Math., 155, 103–147 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Payne, L. E., Sattinger, D. H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math., 22, 273–303 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Rothschild, L. P., Stein, E. M.: Hypoelliptic differential operators and nilpotent groups. Acta Math., 137, 247–320 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Sattinger, D. H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal., 30, 1481–172 (1968)MathSciNetCrossRefGoogle Scholar
  33. [33]
    Schuss, Z.: Theory and Application of Stochastic Differential Equations, Wiley, New York, 1980zbMATHGoogle Scholar
  34. [34]
    Showalter, R. E., Ting, T. W.: Pseudoparabolic partial differential equations. SIAM J. Math. Anal., 1, 1–26 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Sobolev, S. L.: On a new problem of mathematical physics. Izv. Akad. Nauk SSSR Ser. Math., 18, 3–50 (1954)MathSciNetzbMATHGoogle Scholar
  36. [36]
    Ting, T. W.: Parabolic and pseudo-parabolic partial differential equations. J. Math. Soc. Japan, 21, 440–453 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Xu, C. J.: Subelliptic variational problems. Bull. Soc. Math. France, 118, 147–169 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Xu, C. J.: Semilinear subelliptic equations and Sobolev inequality for vector fields satisfying Hörmander’s condition. Chinese J. Contemp. Math., 15, 185–192 (1994)zbMATHGoogle Scholar
  39. [39]
    Xu, R. Z., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal., 264, 2732–2763 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Yung, P. L.: A sharp subelliptic Sobolev embedding theorem with weights. Bull. London Math. Soc., 47, 396–406 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanP. R. China

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