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

TL;DR

This paper proves the global existence of smooth solutions for a class of 3-D quasilinear wave equations with small initial data when a specific nonlinear term vanishes, completing the understanding of blowup versus global existence.

Contribution

It establishes the global existence of solutions in the case where a key nonlinear term is identically zero, complementing previous results on finite-time blowup.

Findings

Global solutions exist when the nonlinear term vanishes.

Previous work showed blowup when the nonlinear term is non-zero.

Complete classification of small data solutions for the equation.

Abstract

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 their arguments. Without loss of generality, one can write $g^{ij}(u, \p u)=c^{ij}+d^{ij}u+\ds\sum_{k=0}^3e^{ij}_k\p_ku+O(|u|^2+|\p u|^2)$, 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 blowup of 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 global existence of the smooth solution $u$. Therefore, the complete results on the blowup or global existence of the small data solutions have been established for the general 3-D quasilinear wave equations $\ds\sum_{i,j=0}^3g^{ij}(u, \p u)\p_{ij}^2u=0$.