arXiv subjects
Publications and source records attributed to Ruying Xue.
The scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation we prove the existence of the scattering operator.
Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2α}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq α\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For $s>-\min\{\frac {3+2α}4, 1\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\frac 12\leα\le 1$ and for $s<-\min\{\frac {3+2α}4, 1\}$ we show some ill-posedness issues.