Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity
The Cauchy problem for the nonlinear dispersive equation of the Korteweg-de Vries type with weak nonlinearity is studied, namely the equation \(u_t+\partial _xf(u)+Ku=0\) with the Cauchy data \(u(0,x)=u_0(x)\). Here, \(K\) is a pseudodifferential operator defined via the Fourier transformation. A particular type of nonlinearities \(f\) with polynomial growth is studied. For sufficiently small (in \(H^{1,1}\) norm) data \(u_0\) the existence of a unique global solution in \(C(\mathbb R,H^{1,1})\) is proven together with the appropriate estimates of Lebesgue norms of \(u\) and \(uu_x\). Moreover, for large time \(t\), the asymptotic formula of the behavior of \(u\) (uniformly in \(x\)) is derived. The technique of Sobolev, weighted Sobolev and Besov spaces is used throughout the proofs, which are split into several, rather technical lemmas. The result is interesting and the paper is, in spite of its technicality, clearly written.
scientific article; zbMATH DE number 1858734