1209.3096 (Changxing Miao et al.)

The Defocusing Energy-Critical Wave Equation with a Cubic Convolution    [PDF]

Changxing Miao, junyong Zhang, Jiqiang Zheng

In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-\Delta u+(|x|^{-4}\ast|u|^2)u=0$ in spatial dimension $d \geq 5$. The main difficulties are the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), which is a fundamental property to exclude the finite time blowup solutions and then to obtain scattering for the wave equations with the local nonlinearity $f(u)=|u|^{2^*-2}u$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. The proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solution; low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the low bound of $\|u\|_{L^{2^\ast}_x}$ to preclude the last two enemies. Finally, we adopt the method of partition timespace norm by time interval in Tao\cite{T07} and the inverse Sobolev inequality to kill the finite time blowup solutions.

View original: http://arxiv.org/abs/1209.3096