tailieunhanh - Báo cáo hóa học: " Research Article Existence of Solutions for Hyperbolic System of Second Order Outside a Domain"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence of Solutions for Hyperbolic System of Second Order Outside a Domain | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 489061 15 pages doi 2009 489061 Research Article Existence of Solutions for Hyperbolic System of Second Order Outside a Domain Jie Xin and Xiuyan Sha School of Mathematics and Information Ludong University Yantai Shandong 264025 China Correspondence should be addressed to Jie Xin fdxinjie@ Received 27 June 2008 Accepted 29 April 2009 Recommended by Robert Bob Gilbert We study the mixed initial-boundary value problem for hyperbolic system of second order outside a closed domain. The existence of solutions to this problem is proved and the estimate for the regularity of solutions is given. The application of the existence theorem to elastrodynamics is discussed. Copyright 2009 J. Xin and X. Sha. 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. 1. Introduction This paper is concerned with the exterior problem for hyperbolic system of second order. Let K be a closed domain with smooth boundary in R3 and let the origin belong to K. Consider the following exterior problem for the hyperbolic system of second order 3 d2u - ỵ aijkl t x djdluk bi i 1 2 3 t x e R X R3 K j k l 1 M u 0 x f x dtu 0 x g x u t x 0 x e dK where aijkl t x e CB 0 to X R3 K and b b1 b2 b3 . We assume that aijkl t x satisfies 3 y aijki t x eijeki a E 2 a 0 j k l 1 2 Journal of Inequalities and Applications -i-iT all CT Tn Tn pfrlp TnafrlvpQ T- Íí TAzhprp 3iMi TT -I- rlTT. IFI2 X 13. .z 2 tor an symmeinc n x ij vviieie eij -1 yui i-jxj uLi-j I jxif xj ij 1eij t x eR XR3 K. Let v dtu. The system can be written as an evolution system in the form dU A t U B at vhere U u1 u2 u3 dtu1 dtu2 õtu3 T u v T B 0 b T . 0 A f at 3 y aijkidjdi Ikawa considered in 1 the mixed problem of a hyperbolic equation of .