一个非齐次抛物方程的有限时间爆破
Finite time blow-up for an inhomogeneous parabolic equation
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中文总结 AI 辅助
研究非齐次抛物方程\(\partial_t u - \Delta u = |u|^{p - 1}u + f(x)\)，构造余维数为\(n\)的非径向初始数据Lipschitz流形，证明其解有限时间爆破且重缩放剖面收敛，表明有限余维稳定性机制在添加非齐次源项时稳健。
中文摘要 AI 辅助
我们考虑非齐次非线性热方程\(\partial_t u - \Delta u = |u|^{p - 1}u + f(x)\)，其中\(x\in\mathbb{R}^3\)，\(p > 5\)，且\(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\)。对于每个足够大的整数\(n\)，我们构造了一个余维数为\(n\)的非径向初始数据的Lipschitz流形，其相应的解在有限时间内爆破，并且其重缩放剖面收敛到齐次方程规定的自相似剖面\(\Phi_n\)。主要新颖之处在于表明，在Collot、Raphaël和Szeftel [《美国数学会论文集》(2019)]中为齐次方程发展的自相似爆破的有限余维稳定性机制，在添加有界、Lipschitz空间非齐次源项的情况下是稳健的。与齐次问题不同，这里考虑的方程没有精确的尺度不变性。我们期望这项工作中发展的框架也可能对精确尺度不变性被破坏的相关问题有用。
英文摘要
We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-Δu=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where \(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\). For every sufficiently large integer \(n\), we construct a codimension-\(n\) Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile \(Φ_n\) of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Raphaël and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.
