临界wave equation的非散射解和无穷远爆破
Nonscattering solutions and blowup at infinity for the critical wave equation
摘要：
本文在$\R^{1+3}$中考虑临界聚焦波方程$(-\partial_t^2+Δ)u+u^5=0$，证明了在前向光锥${(t,x)\in\R\times \R^3: |x|\leq t, t\gg 1}$中存在形式为[u（t，x）=t^ \frac μ{2}W（t^μx）+η（t，x）]的能量类解，其中$W(x)=(1+(1/3)|x|^2)^{-(1/2)}$为基态孤子，$μ$为任意给定的实数（正或负），且误差$η$满足[|\partial_t η（t，\cdot）|_{L^2（B_t）} +| \nabla η（t，\cdot）|_{L^2（B_t）} \ll 1，\quad B_t：={x \in \R^3：|x| <t}]。此外，$u$在圆锥外的动能很小。因此，根据符号的$μ$，我们得到两种新类型的解决方案，要么集中为$t\to\infty$（与连续的速率）或保持有界，但不分散。特别是，这些解决方案相矛盾的一个强版本的孤子决议猜想。
英文摘要：
We consider the critical focusing wave equation $(-\partial_t^2+Δ)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form [u(t,x)=t^\fracμ{2}W(t^μx)+η(t,x)] in the forward lightcone ${(t,x)\in\R\times \R^3: |x|\leq t, t\gg 1}$ where $W(x)=(1+(1/3)|x|^2)^{-(1/2)}$ is the ground state soliton, $μ$ is an arbitrary prescribed real number (positive or negative) with $|μ|\ll 1$, and the error $η$ satisfies [|\partial_t η(t,\cdot)|_{L^2(B_t)} +|\nabla η(t,\cdot)|_{L^2(B_t)}\ll 1,\quad B_t:={x\in\R^3: |x|<t}] for all $t\gg 1$. Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $μ$, we obtain two new types of solutions which either concentrate as $t\to\infty$ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
