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

The main purpose of the present paper is to study the blow-up problem of thewave equation with space-dependent damping in the \textit{scale-invariant case}and time derivative nonlinearity with small initial data. Under appropriateinitial data which are compactly supported, by using a test function method andtaking into account the effect of the damping term($\frac{\mu}{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions theblow-up region is given by $p \in (1, p_G(N+\mu)]$ where $p_G(N)$ is theGlassey exponent. Furthermore, we shall establish a blow-up region, independentof $\mu$ given by $p\in (1, 1+\frac{2}{N}),$ for appropriate initial data inthe energy space with noncompact support.

Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture