ON THE DECAY OF STEP FOR THE KORTEWEG-DEVRIES-BURGERS EQUATION

In [1] the question of the decay of step-like data for the Korteweg-de Vries-Burgers equation u(t) + 2uu(x) + au(xxx) - u(xx) = 0 in the case Absolute value of a less-than-or-equal-to 1/8 was considered, and it was proved that the solution of the Cauchy problem for Eq. (1) tends uniformly in x, as t --> infinity, to the stationary wave phi(x) defined by aphi(xx) - phi(x) + phi2 - 1 = 0, phi\x-->+/-infinity = -/+1. (2) In [2] it was shown that the solution phi(x) of the problem (2) exists and is determined uniquely up to a shift of the argument. We fix this possible shift by the condition phi(0) = 0. In [2] it was also proved that in the case Absolute value of a less-than-or-equal-to 1/8 the solution phi(x) of (2) decreases monotonically, i.e., phi'(x) < 0 for all x is-an-element-of R1 , and in the case Absolute value of a > 1/8, the function phi(x) decreases monotonically for x greater-than-or-equal-to x0 and oscillates for x < x0, where x0 is the maximum point of phi(x). In this paper we study the behavior, as t --> infinity, of the solution u(x, t) of the Cauchy problem for Eq. (1) in the case Absolute value of a > 1/8. We prove that the solution u(x, t) converges to the stationary wave phi(x) as t --> infinity uniformly with respect to x is-an-element-of R1 in the case Absolute value of a > 1/8, and we obtain an estimate for the rate of convergence. Thus, the stationary wave is also asymptotically stable in the case Absolute value of a > 1/8.