论文标题
在$ \ mathbb {r}^{1+2} $中使用半线性波方程的全局行为
On the global behaviors for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$
论文作者
论文摘要
在本文中，我们研究了$ \ mathbb {r}^{1+2} $以纯电源非线性散发半连接波方程的渐近衰减属性。通过将新的矢量字段应用于空的超平面，我们得出了势能的改善时间衰减，结果是，对于所有$ p> 1+ \ sqrt {8} $，解决方案既散布在关键的Sobolev空间和能量空间中。此外，与Brézis-Gallouet-Wainger类型的对数Sobolev嵌入类型相结合，我们表明该解决方案在$ p>> \ frac {11}} {3} {3} {3} {3} $ t和所有费率的$ t^frac+for $ t^forac+for $ t^$} $} $} $} $} $ {po} $} $ {po} $} $} $ {po} $} $} $} $ {po} $} $ {po} $} $} $ {p时{p i} $ {po} $ 1 <p \ leq \ frac {11} {3} $。这特别意味着当$ p> 2 \ sqrt {5} -1 $时，解决方案会散布在能量空间中。
In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+\sqrt{8}$. Moreover combined with Brézis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-\frac{1}{2}}$ when $p>\frac{11}{3}$ and with rate $t^{ -\frac{p-1}{8}+ε}$ for all $1<p\leq \frac{11}{3}$. This in particular implies that the solution scatters in energy space when $p>2\sqrt{5}-1$.
