论文标题
非线性快速扩散在有界域上的渐近学接近灭绝
Asymptotics near extinction for nonlinear fast diffusion on a bounded domain
论文作者
论文摘要
在平滑的有界欧几里得域上，已知Sobolev-屈服的快速扩散会导致有限的时间灭绝，而初始基准选择了消失的轮廓。在重新验证的变量中，我们以相对误差均匀地量化了与该轮廓的收敛速率，这表明速率要么快速快（由频谱差距预测的速率常数），要么是代数缓慢（仅在存在不可整合零模式的情况下才有可能）。在第一种情况下，非线性动力学通过指数衰减的特征模式至少是间隙的两倍，可以很好地X型。这是完善并确认了1980年的贝里曼和荷兰的猜想。我们还通过提供一种新的，更简单的方法来适应零模式的存在，例如当消失的曲线未能隔离时发生的方法（可能属于此类概况的连续体），我们还可以改善Bonforte和Figalli的结果。
On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).
