论文标题
在$ \ mathbb {r}^{1+1} $中脱落半线性波方程的渐近衰变
Asymptotic decay for defocusing semilinear wave equations in $\mathbb{R}^{1+1}$
论文作者
论文摘要
本文致力于研究溶液的渐近行为，以实现一维偏置的半线性波方程。我们证明，有限的能量解决方案在刻有意义上趋于零，从而改善了Lindblad和Tao的平均衰变。此外，对于属于某些加权能量空间的足够局部数据，该解决方案会以逆多项式速率及时衰减。这证实了上述工作中提出的猜想。 结果基于新的加权矢量场，作为将乘数应用于光线封闭的区域的乘数。第一个结果的关键观察结果是势能的局部能量衰减，而第二个结果依赖于一种加权的Gagliardo-Nirenberg不平等。
This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.
