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 Exponential Stability of Two Coupled Second-Order Evolution Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 879649 14 pages doi 2011 879649 Research Article Exponential Stability of Two Coupled Second-Order Evolution Equations Qian Wan and Ti-Jun Xiao Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China Correspondence should be addressed to Ti-Jun Xiao xiaotj@ Received 30 October 2010 Accepted 21 November 2010 Academic Editor Toka Diagana Copyright 2011 Q. Wan and . Xiao. 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. By using the multiplier technique we prove that the energy of a system of two coupled second order evolution equations one is an integro-differential equation decays exponentially if the convolution kernel k decays exponentially. An example is give to illustrate that the result obtained can be applied to concrete partial differential equations. 1. Introduction Of concern is the exponential stability of two coupled second-order evolution equations one is an integro-differential equation in Hilbert space H t u if Au if au if -J k t - s Au s ds f VAv if 0 v t Avft f VAu t 0 with initial data u 0 u0 v 0 v0 u 0 u1 v 0 v1. Here A D A c H H is a positive self-adjoint linear operator a 0 Ộ 0 k f is a nonnegative function on 0 to . Moreover the fractional power A1 2 is defined as in the well known operator theory cf . 1 2 . 2 Advances in Difference Equations An interesting and difficult point for it is to stabilize the whole system via the damping effect given by only one equation . We remark that there is very few work concerning the situation when the damping mechanism is given by memory terms see 3 where a coupled Timoshenko beam system is investigated. On the other hand the stability of the single
