论文标题
3D能量亚临界波方程的弱非辐射径向溶液
Weakly non-radiative radial solutions to 3D energy subcritical wave equations
论文作者
论文摘要
在这项工作中，我们考虑能量亚临界3D波方程$ \ partial_t^2 u-Δu= \ pm | u | u |^{p-1} u $并讨论其（弱）非辐射解解决方案，即在外部区域中定义的解决方案$ \ \ \ {（x，x，t） \ [ \ lim_ {t \ rightarrow \ pm \ infty} \ int_ {| x |> |> | t | + r} \ left（| \ nabla u（x，x，x，t）|^2 + | u_t（x，x，t）|^2 \ right）dx = 0。 \]众所周知，对线性波方程的任何径向弱的非辐射解是$ 1/| x | $的倍数。此外，对于能量关键波方程的任何径向弱非辐射解决方案$ u $都必须具有类似的渐近行为，即$ u（x，x，t）\ simeq c/| x | $当$ | x | $很大。在这项工作中，我们举例说明，径向弱的非辐射式解决方案（$ 3 <p <5 $）可能具有差异的渐近行为。但是，一个径向弱的非辐射解决方案$ u $在关键的sobolev空间中具有初始数据$ \ dot {h}^{s_p} \ times \ dot \ dot {h}^{s_p-1}（s_p-1}（\ mathbb {r}^3） - | w |^{p -1} w $，以便$ u（x，t）\ equiv w（x）\ simeq c/| x | $当$ | x | $很大。
In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - Δu = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with $R\geq 0$ satisfying \[ \lim_{t\rightarrow \pm\infty} \int_{|x|>|t|+R} \left(|\nabla u(x,t)|^2 + |u_t(x,t)|^2\right) dx = 0. \] It has been known that any radial weakly non-radiative solution to the linear wave equation is a multiple of $1/|x|$. In addition, any radial weakly non-radiative solutions $u$ to the energy critical wave equation must possess a similar asymptotic behaviour, i.e. $u(x,t)\simeq C/|x|$ when $|x|$ is large. In this work we give examples to show that radial weakly non-radiative solutions to energy subcritical equation ($3<p<5$) may possess a much different asymptotic behaviour. However, a radial weakly non-radiative solution $u$ with initial data in the critical Sobolev space $\dot{H}^{s_p}\times \dot{H}^{s_p-1}(\mathbb{R}^3)$ must coincide with a $C^2$ solution $W$ to the elliptic equation $-ΔW = -|W|^{p-1} W$ so that $u(x,t) \equiv W(x) \simeq C/|x|$ when $|x|$ is large.
