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 consider the wave equation with variable coefficients on an exteriordomain in $\R^n$($n\ge 2$). We are interested in finding a special uniformdecay rate of local energy different from the constant coefficient waveequation.  More concretely, if the dimensional $n$ is even, whether the uniform decayrate of local energy for the wave equation with variable coefficients can breakthrough the limit of polynomial and reach exponential; if the dimensional $n$is odd, whether the uniform decay rate of local energy for the wave equationwith variable coefficients can hold exponential as the constant coefficientwave equation .  \quad \ \ We propose a cone and establish Morawetz's multipliers in a versionof the Riemannian geometry to derive uniform decay of local energy for the waveequation with variable coefficients. We find that the cone with polynomialgrowth is closely related to the uniform decay rate of the local energy. Moreconcretely, for radial solutions, when the cone has polynomial of degree$\frac{n}{2k-1}$ growth, the uniform decay rate of local energy is exponential;when the cone has polynomial of degree $\frac{n}{2k}$ growth, the uniform decayrate of local energy is polynomial at most. In addition, for general solutions,when the cone has polynomial of degree $n$ growth, we prove that the uniformdecay rate of local energy is exponential under suitable Riemannian metric. Itis worth pointing out that such results are independent of the parity of thedimension $n$, which is the main difference with the constant coefficient waveequation. Finally, for general solutions, when the cone has polynomial ofdegree $m$ growth, where $m$ is any positive constant, we prove that theuniform decay rate of the local energy is of primary polynomial under suitableRiemannian metric.

Special uniform decay rate of local energy for the wave equation with variable coefficients on an exterior domain