English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we consider the following Cauchy problem of \begin{equation*}\left\{ \begin{array}{lll} iu_t=\Delta u+2\delta_huh'(|u|^2)\Deltah(|u|^2)+V(x)u+F(|u|^2)u+(W*|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here$\delta_h$ is a constant, $N\geq 1$, $h(s)$, $F(s)$, $V(x)$ and $W(x)$ are somereal functions, $W(x)$ is even. Besides obtaining some sufficient conditions onglobal existence of the solution, we establish pseudoconformal conservation lawand give Morawetz type estimates, spacetime bounds and asymptotic behaviors forthe global solution. We bring two ideas to establish scattering theory, one isthat we take different admissible pairs in Strichartz estimates for differentterms on the right side of Duhamel's formula in order to keep each termindependent, another is that we factitiously let a continuous function be thesum of two piecewise functions and chose different admissible pairs inStrichartz estimates for the terms containing these functions. Basing on thetwo ideas, we provide the direct and simple proofs of classic scatteringtheories in $L^2(\mathbb{R}^N)$ and $\Sigma$ for any space dimension($N\geq 1$)under certain assumptions. Here $$ \Sigma=\{u\in H^1(\mathbb{R}^N),\quad|xu|\in L^2(\mathbb{R}^N)\}. $$

Spacetime estimates and scattering theory for quasilinear Schr\"{o}dinger equations in arbitrary space dimension