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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We establish local well-posedness results for the Initial Value Problemassociated to the Schr\"odinger-Debye system in dimensions $N=2, 3$ for data in$H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell\le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\timesL^2$. Our results improve the previous ones obtained in \cite{Bidegaray1},\cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case(N=2) and for initial data in $H^1\times L^2$, we prove that solutions existfor all times, thus providing a negative answer to the open problem mentionedin \cite{Fibich-Papanicolau} concerning the formation of singularities forthese solutions.

Local and Global Well-Posedness for the Critical Schrodinger-Debye System