Title: The radiation theory of radial solutions to 3D energy critical wave equations Show Chinese title

Title: 径向解的辐射理论对三维能量临界波动方程

Abstract: In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave equation is similar to that of a linear free wave $v_L$ in an exterior region $\{x: |x|>R+|t|\}$, i.e. \[ \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 + |u_t-\partial_t v_L|^2) dx = 0. \] We classify all such solutions for a given linear free wave $v_L$ in this work. We also give some applications of our theory on the global behaviours of radial solutions to this kind of equations. In particular we show the scattering of all finite-energy radial solutions to the defocusing energy critical wave equations. Show Chinese abstract

Abstract: 在本工作中，我们考虑三维空间中具有径向数据的广泛能量临界波动方程。我们感兴趣的是外部散射现象，在这种现象中，解$u$对非线性波动方程的渐近行为类似于线性自由波$v_L$在外部区域$\{x: |x|>R+|t|\}$中的行为，即\[ \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 + |u_t-\partial_t v_L|^2) dx = 0. \]。我们在本工作中对给定的线性自由波$v_L$对所有此类解进行分类。我们还给出了我们的理论在该类方程的径向解全局行为上的某些应用。特别是，我们展示了所有有限能量的径向解在非聚焦能量临界波动方程中的散射。