Acta Mathematica Sinica, English Series

, Volume 34, Issue 6, pp 959–974 | Cite as

One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources

  • Yang Cao
  • Jing Xue Yin
  • Ying Hua Li


In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T > 0, the solution will never blow up during the period 0 < t < T, so long as the viscous coefficient is large enough.


Cahn–Hilliard pseudo-parabolic asymptotic behavior 

MR(2010) Subject Classification

35A01 35B40 35K35 


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianP. R. China
  2. 2.School of Mathematical SciencesSouth China Normal UniversityGuangzhouP. R. China

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