Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified space domain. Blow up in finite time is caused by these singularities eventually reaching the real axis. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales on which blow up is approached. It is shown that an ordinary differential equation with quadratic nonlinearity plays a central role in the asymptotic analysis. This equation is studied in detail, including its numerical computation on multiple Riemann sheets, and the far-field solutions are shown to be given at leading order by a Weierstrass elliptic function.

翻译：通过渐近分析和多种数值方法，研究了具有空间周期性和二次非线性的热方程的爆破解。研究重点在于复空间域中奇点的动力学行为。有限时间内的爆破是由这些奇点最终到达实轴引起的。分析揭示了小非线性效应与大非线性效应之间的区别，并提供了爆破接近过程中不同时间尺度的洞察。结果表明，一个具有二次非线性的常微分方程在渐近分析中起核心作用。对该方程进行了详细研究，包括其在多个黎曼曲面上的数值计算，并证明远场解的首阶近似由魏尔斯特拉斯椭圆函数给出。