Abstract
We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg-de Vries equation u/sub t/+a(t)(u/sup 3/)/sub x/+1/3u/sub xxx/=0,(t,x)/spl isin/R/spl times/R, with initial data u(0,x)=u/sub 0/(x),x/spl isin/R. We assume that the coefficient a(t)/spl isin/C/sup 1/(R) is a real, bounded and slowly varying function, such that |a'(t)|/spl les/C(1+|t|)/sup -7/6/. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space. We prove the time decay estimates of the solutions. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.
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