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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 568404, 12 pages
http://dx.doi.org/10.1155/2012/568404
Research Article

The Local Strong and Weak Solutions for a Generalized Pseudoparabolic Equation

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 7 February 2012; Accepted 21 March 2012

Academic Editor: Yonghong Wu

Copyright © 2012 Nan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space 𝐶 ( [ 0 , 𝑇 ) ; 𝐻 𝑠 𝐶 ( 𝑅 ) ) 1 ( [ 0 , 𝑇 ) ; 𝐻 𝑠 1 ( 𝑅 ) ) with 𝑠 > 3 / 2 , while the existence of local weak solutions is proved in the space 𝐻 𝑠 ( 𝑅 ) with 1 𝑠 3 / 2 . Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space 𝐶 ( [ 0 , ) ; 𝐻 𝑠 𝐶 ( 𝑅 ) ) 1 ( [ 0 , ) ; 𝐻 𝑠 1 ( 𝑅 ) ) with 𝑠 2 .