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Indexed by:期刊论文

Date of Publication:2018-06-01

Journal:ACTA MATHEMATICA SINICA-ENGLISH SERIES

Included Journals:SCIE

Volume:34

Issue:6

Page Number:959-974

ISSN No.:1439-8516

Key Words:Cahn-Hilliard; pseudo-parabolic; asymptotic behavior

Abstract:In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Phi(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 Phi(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.