Large-time Behavior of Solutions of Burgers' Equation

The large-time asymptotic behavior of real-valued solutions of the pure initial-value problem for Burgers' equation u t + uu x ? u xx = 0, > 0, is studied. The initial data is of the form u 0 (x) = nx 1+x 2 + u 1 (x), where n 2 R and u 1 2 L 1 (R). Eberhard Hopf H] considered the case n = 0, and the case ku 1 k L 1 == + jnj suuciently small was considered in D]. Here we study the general case. When n < 1 we nd an explicit function U() such that u(x; t) = t ?1=2 U(xt ?1=2) + o(t ?1=2 (1 + jxjt ?1=2) ?1) as t ! 1. When n 1 the solution is no longer asymptotically self-similar, although it almost is. We give a matched asymptotic description of the solution and compute its sharp decay rate in L 1 .