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 are concerned with the 3-D quasilinear wave equation $\ds\sum_{i,j=0}^3g^{ij}(u, \p u)\p_{ij}^2u$ $=0$ with $(u(0,x),\p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $x_0=t$, $x=(x_1, x_2, x_3)$,$\p=(\p_0, \p_1, ..., \p_3)$, $u_0(x), u_1(x)\in C_0^\infty(\Bbb R^3)$, $\ve>0$is small enough, and $g^{ij}(u, \p u)=g^{ji}(u, \p u)$ are smooth in theirarguments. Without loss of generality, one can write $g^{ij}(u, \pu)=c^{ij}+d^{ij}u+\ds\sum_{k=0}^3e^{ij}_k\p_ku+O(|u|^2+|\p u|^2)$, where$c^{ij}, d^{ij}$ and $e^{ij}_k$ are some constants, and$\ds\sum_{i,j=0}^3c^{ij}\p_{ij}^2=-\square\equiv -\p_t^2+\Delta$. When$\ds\sum_{i,j,k=0}^3e^{ij}_k\o_k\o_i\o_j\not\equiv 0$ for $\o_0=-1$ and$\o=(\o_1, \o_2, \o_3)\in\Bbb S^2$, the authors in [7-8] have shown the blowupof the smooth solution $u$ in finite time as long as $(u_0(x),u_1(x))\not\equiv 0$. In the present paper, when$\ds\sum_{i,j,k=0}^3e^{ij}_k\o_k\o_i\o_j\equiv 0$, we will prove the globalexistence of the smooth solution $u$. Therefore, the complete results on theblowup or global existence of the small data solutions have been establishedfor the general 3-D quasilinear wave equations $\ds\sum_{i,j=0}^3g^{ij}(u, \pu)\p_{ij}^2u=0$.

Global smooth solutions of 3-D quasilinear wave equations with small initial data