Journal of Applied Mathematics
Volume 2012 (2012), Article ID 805158, 15 pages
http://dx.doi.org/10.1155/2012/805158
Research Article

Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

Zhigang Pan,1 Hong Luo,2 and Tian Ma1

1Yangtze Center of Mathematics, Sichuan University, Chengdu, Sichuan 610041, China
2College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China

Received 20 February 2012; Revised 8 May 2012; Accepted 9 May 2012

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2012 Zhigang Pan et al. 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

We consider the global existence of strong solution 𝑢 , corresponding to a class of fully nonlinear wave equations with strongly damped terms 𝑢 𝑡 𝑡 − 𝑘 Δ 𝑢 𝑡 = 𝑓 ( 𝑥 , Δ 𝑢 ) + 𝑔 ( 𝑥 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) in a bounded and smooth domain Ω in 𝑅 𝑛 , where 𝑓 ( 𝑥 , Δ 𝑢 ) is a given monotone in Δ 𝑢 nonlinearity satisfying some dissipativity and growth restrictions and 𝑔 ( 𝑥 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) is in a sense subordinated to 𝑓 ( 𝑥 , Δ 𝑢 ) . By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution 𝑢 ∈ 𝐿 ∞ l o c ( ( 0 , ∞ ) , 𝑊 2 , 𝑝 ( Ω ) ∩ 𝑊 0 1 , 𝑝 ( Ω ) ) .