Decay Behavior Of Nonlinear Evolution Equation

This paper is mainly concerned with decay behavior and the existence of the global solution of Cauchy problem for several kinds of nonlinear evolution equations.First of all, we study the the decay rate,upper and lower bounds, Optimization for the nonlinear parabolic equation (1) with damping. Further, With the aid of the classic Fourier splitting methods, the solution decays at (1+t) n/4and the solution exists and decays in L2norms C,(1+t)n/4≤||u(x, t)||≤C2(1+t)n/4are proved.,which is the same as the solutions of the heat equation.Next,the existence and uniqueness of the global solution of Cauchy problem for a class of semi-linear heat conduction equation (2) are studied in this paper.Utilizing the estimation of energy and the estimation of degeneration together,and by the Banach fixed point theorem,the existence and uniqueness of the global solution are presented,sa the initial value φ and the non-linear term ur-bu satisfy certain conditions,then,we discuss the decay estimate of the large time behavior for the semi-linear heat conduction equation.Finally,by using the Galerkin’s method,Banach’s Fixed Piont Theorem and contract mapping principle,we prove the existence and uniqueness and decay of global solutions of the initial value problem for the n-dimensional nonlinear wave equation (3).