The Exponential Decay For Solutions To Two Non-linear Evolution Equation

In this paper,we study the exponential decay for solutions to two non-linear evolution equation in mathematics and physics.In the second chapter, the exponential decay for solutions to Kirchhoff equation is stud-ied based on method of nonlinear kirchhoff equation and nonlinear wave equation.The boundedness of solutions is obtained by Galerkin approxima-tion.Moreover,exponentially decay of solution for Kirchhoff equation are proved at specific condition by establishing an appropriate Lyapunov function.The method play a positive role for further study of Kirchhoff equation.In the third chapter;it is concerned with the exponential decay for solutions to KDV equation with localized damping.Combing multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions.