Decay Estimate Of The Coupled System Of Second Order Evolution Equations

This thesis investigates a system of two coupled second order evolution equations (one has a memory term) in a Hilbert space. It is divided into four parts. The first part introduces the background and some historical research of this problem. In the second part, we rewrite the problem as a single equation by using variable substitution and get the local existence of the solution from the known facts. Then, we prove that the related terms are uniformly bounded and show the global existence of the solution. In the third part, we use the multiplier technique to get an integral inequality about the system energy. There is a term which is called memory-energy on the right of the inequality. We estimate this term under several assumptions and get the decay estimate of the energy. The final part applies our abstract results to two concrete problems and gives some notes about our assumptions and results.