Asymptotic behavior in time of the solutions of three nonlinear partial differential equations

Asymptotic behavior in time of the solutions of three nonlinear partial differential equations. Consider the following three equations with space dimension one: 1. (1) ut + (uxx − up − u)x = 0, where p ⩾ 3 is an odd integer, 2. (2) utt − uxx + (g(x) + 1)up + g(x)u = 0, where p ⩾ 3 is an odd integer, gx 0, and g and all its derivatives up to order two are bounded, 3. (3) iut − uxx + ¦ u ¦2pu + k(x)u = 0, where p is a positive integer, k > 0, k and all its derivatives up to order three are bounded, and there is a real-valued function A of x such that Axxx + 2Akx 0, and ¦A¦, ¦Ax¦, and ¦Axx¦ are bounded. (Example of k: k(x) = 1(1 + a2x2) with 0 < a ⩽ (23)12.) We prove that the local L2 norms of the smooth solutions of the above three equations with nice initial data decay to zero for large time.