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Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations

Published online by Cambridge University Press:  14 August 2017

Xiaoli Zhu
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, People's Republic of China (yhli@sxu.edu.cn)
Fuyi Li
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, People's Republic of China (yhli@sxu.edu.cn)
Yuhua Li*
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, People's Republic of China (yhli@sxu.edu.cn)
*
*Corresponding author.
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In this paper we are interested in a sharp result about the global existence and blowup of solutions to a class of pseudo-parabolic equations. First, we represent a unique local weak solution in a new integral form that does not depend on any semigroup. Second, with the help of the Nehari manifold related to the stationary equation, we separate the whole space into two components S+ and S via a new method, then a sufficient and necessary condition under which the weak solution blows up is established, that is, a weak solution blows up at a finite time if and only if the initial data belongs to S. Furthermore, we study the decay behaviour of both the solution and the energy functional, and the decay ratios are given specifically.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2017