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


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.


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

MR(2010) Subject Classification

35K55 35K65 


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


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© 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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