半线性热方程的振荡爆破与梯度估计
Oscillatory blow-up and gradient estimates for semilinear heat equations
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中文总结 AI 辅助
针对半线性热方程爆破理论的基础问题，构造了任意超临界幂次下具有振荡L^∞范数的非径向爆破反例，扩展了II型爆破的幂次范围并明确了梯度估计的适用时刻。
中文摘要 AI 辅助
对于具有爆破非线性项的反应扩散问题，我们研究任意正爆破解的上确界范数是否最终必须随时间单调非递减的问题。尽管已有一些充分条件，特别是针对径向解的情况，但爆破理论中这一自然且基本的问题迄今似乎尚未得到完全一般性的解决。我们针对任意索伯列夫超临界幂次非线性项，构造了具有振荡L^∞范数的（非径向）爆破解的反例，表明该性质可能不成立。此外，这为任意超临界幂次提供了II型爆破的例子，大幅扩展了II型爆破可能发生的已知幂次范围。而且，此前已知的所有II型爆破速率至多为多项式，而我们反例中的爆破可以具有任意奇异性。作为相关问题，我们阐明了前人工作中得到并使用的梯度估计，特别证明这些估计仅在L^∞范数相对于过去取最大值的时刻成立。
英文摘要
For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L^\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L^\infty$ norm is maximal with respect to the past.
