论文标题
与光锥超出二次波方程的共同尺寸稳定的爆炸
Co-dimension one stable blowup for the quadratic wave equation beyond the light cone
论文作者
论文摘要
我们研究了最低能量超临界尺寸$ d = 7 $的二次波方程中明显已知的非平凡的自相似爆炸解决方案的稳定性。该解决方案在一个点上吹来，并自然远离奇异性。通过使用倍或相似性坐标，我们证明了该溶液的条件非线性渐近稳定性在时空区域中的小，紧凑的径向扰动下，可以任意地将其与奇异性的凯奇（Cauchy）地平线保持联系。为了实现这一目标，我们严格解决了基本的光谱问题，并表明该解决方案完全不稳定。该解决方案的不稳定性质需要仔细构建$ t = 0 $的适当调整的初始数据，当该数据传播到一个恒定倍重时间的空格性超丘家族时，它采取了所需的形式，以确保收敛。通过这种情况，我们引入了一种新的规范方法，以研究在倍boid型相似性坐标框架内针对非线性波方程的不稳定的自相似解决方案。
We study the stability of an explicitly known, non-trivial self-similar blowup solution of the quadratic wave equation in the lowest energy supercritical dimension $d = 7$. This solution blows up at a single point and extends naturally away from the singularity. By using hyperboloidal similarity coordinates, we prove the conditional nonlinear asymptotic stability of this solution under small, compactly supported radial perturbations in a region of spacetime which can be made arbitrarily close to the Cauchy horizon of the singularity. To achieve this, we rigorously solve the underlying spectral problem and show that the solution has exactly one genuine instability. The unstable nature of the solution requires a careful construction of suitably adjusted initial data at $t = 0$, which, when propagated to a family of spacelike hypersurfaces of constant hyperboloidal time, takes the required form to guarantee convergence. By this, we introduce a new canonical method to investigate unstable self-similar solutions for nonlinear wave equations within the framework of hyperboloidal similarity coordinates.
