The Research Of Decays Of Solutions For Viscoelastic Equation (Equations) With Nonlinear Boundary Conditions

Nowadays, with the development of information technology and rapid progress of sci-ence and technology, many natural phenomena concerned with space have been analysed and paid more attention to, and those natural phenomena can always been abstracted into nonlinear partial differential equation, so the study on partial differential equation has be-came a significant subject in the field of mathematics. Besides, the phenomena of nonlinear vibration and nonlinear viscoelastic vibration can always been found in physics, and the researches on those phenomena are of great importance in real life, and also has made a great progress.This thesis mainly investigate decays of quasilinear viscoelastic equation with nonlinear boundary conditions and a system of two quasilinear viscoelastic equations with coupled non-linear boundary conditions. The first problem which this thesis investigate is a quasilinear viscoelastic equation with initial values and two compatible conditions, we obtain the decay of the solution, and the second one is the decay of solutions for a system of two quasilinear viscoelastic equations with coupled nonlinear boundary conditions, and this thesis conclude that the dissipations given by the viscoelastic terms are strong enough to ensure the decay of the solutions energy according to differential inequalities and perturbed energy method, furthermore the decay are based on the property of the relaxation function.