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。 II，冲击形成
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. II, shock formation
论文作者
论文摘要
在上一篇论文[ding bingbing，lu yu，yin huicheng上，在3D Quasilinear Wave方程的关键指数上$ P_C $ $ - \ big（1+（\ partial_tϕ）^p \ big）\ partial_t^p partial_t^2ϕ+δϕ = 0 $具有短脉冲初始数据。 i，全局存在，预印度，2022]，对于3D quasilinear Wave方程$ - \ big（1+（\ partial_tx）^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） $p\in\mathbb{N}$, $0<\varepsilon_0<1$, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0-k\max\{0,1-(1-\varepsilon_0)p\}})$对于$ k = 1,2 $，当$ p> p_c $带有$ p_c = \ frac {1} {1- \ varepsilon_0} $时，作者建立了平滑大解决方案$ ϕ $的全局存在。在本文中，在相同的传出约束条件下，当$ 1 \ leq p \ leq p_c $时，我们将证明平滑的解决方案$ ϕ $爆炸了，进一步在有限的时间内形成了即将发出的冲击。
In the previous paper [Ding Bingbing, Lu Yu, Yin Huicheng, 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, Preprint, 2022], for the 3D quasilinear wave equation $-\big(1+(\partial_tϕ)^p\big)\partial_t^2ϕ+Δϕ=0$ with 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 $p\in\mathbb{N}$, $0<\varepsilon_0<1$, under the outgoing constraint condition $(\partial_t+\partial_r)^kϕ(1,x)=O(δ^{2-\varepsilon_0-k\max\{0,1-(1-\varepsilon_0)p\}})$ for $k=1,2$, the authors establish the global existence of smooth large solution $ϕ$ when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$. In the present paper, under the same outgoing constraint condition, when $1\leq p\leq p_c$, we will show that the smooth solution $ϕ$ blows up and further the outgoing shock is formed in finite time.
