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 show almost global existence of small solutions to theCauchy problem for symmetric system of wave equations with quadratic (in 3D) orcubic (in 2D) nonlinear terms and multiple propagation speeds. To measure thesize of initial data, we employ a weighted Sobolev norm whose regularity indexis the smallest among all the admissible Sobolev norms of integer order. Wemust overcome the difficulty caused by the absence of the $H^1$-$L^p$Klainerman-Sobolev type inequality, in order to obtain a required a prioribound in the low-order Sobolev norm. The introduction of good substitutes forthis inequality is therefore at the core of this paper. Using the idea ofshowing the well-known Lady\v{z}enskaja inequality, we prove some weightedinequalities, which, together with the generalized Strauss inequality, play arole as the good substitute.

Regularity and lifespan of small solutions to systems of quasi-linear wave equations with multiple speeds,I: almost global existence