论文标题
在关键指数上$ p_c $的3D QuasilIrinear Wave方程$ - \ big（1+（\ partial_tϕ）^p \ big）\ partial_t^2ϕ+δϕ = 0 $，带有短脉冲初始数据。我，全球存在
On the critical exponent $p_c$ of the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with short pulse initial data. I, global existence
论文作者
论文摘要
对于3D quasilinear Wave方程$ - \ big（1+（\ partial_tϕ）^p \ big）\ partial_t^2ϕ+δ= 0 $，带有短脉冲初始数据$（ϕ，\ partial_tϕ）（1，x）= \ big（δ^{2- \ varepsilon_0} ϕ_0（\ frac {r-1}δ，ω），Δ^^^^{1- \ \ varepsilon_0} ϕ_1（\ frac） $ p \ in \ mathbb n $，$ p \ geq 2 $，$ 0 <\ varepsilon_0 <1 $，$ r = | x | $，$ω= \ frac {x} {x} {r} {r} \ in \ mathbb s^2 $，$ umatebb s^2 $，以及$δ> 0 $是足够小的。 $（\ partial_t+\ partial_r）^kϕ（1，x）= o（δ^{2- \ varepsilon_0}）$对于$ k = 1,2 $，我们将在$ p_c $ p_c = $ p_c = \ frac = \ frac p_c = \ frac {1} trigite vareps的全球存在$ p> p_c $ p_c $时，我们将建立平滑的大数据解决方案$ ϕ $。在即将发表的论文中，当$ 1 \ leq p \ leq p_c $时，我们在$（ϕ_0，ϕ_1）$的合适假设下显示了在时间$ t = 2 $之前的即将上流电击的形成。
For the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with the short pulse initial data $(ϕ,\partial_tϕ)(1,x)=\big(δ^{2-\varepsilon_0}ϕ_0(\frac{r-1}δ,ω), δ^{1-\varepsilon_0}ϕ_1(\frac{r-1}δ,ω)\big)$, where $p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $ω= \frac{x}{r}\in\mathbb S^2$, and $δ>0$ is sufficiently small, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0})$ for $k=1,2$, we will establish the global existence of smooth large data solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent. In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the outgoing shock before the time $t=2$ under the suitable assumptions of $(ϕ_0,ϕ_1)$.
