We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges equals the sum of the usual dimensional decay rate and a multiple of the reciprocal value of the opening angle. To prove the result, we develop the method of self-similar variables for the associated heat equation and study the asymptotic behaviour of the transformed non-autonomous parabolic problem for large times. We also establish an improved Hardy inequality for the Dirichlet Laplacian in non-trivially curved wedges and state a conjecture about an improved decay rate in this case.

中文翻译：

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哈代不等式和弯曲楔中的热流

我们表明，弯曲平面楔中狄利克雷拉普拉斯算子的热半群的多项式衰减率等于通常的维数衰减率和张角倒数的倍数之和。为了证明结果，我们开发了相关热方程的自相似变量方法，并研究了大时间变换的非自治抛物线问题的渐近行为。我们还为非平凡弯曲的楔形中的狄利克雷拉普拉斯算子建立了改进的哈代不等式，并在这种情况下提出了关于改进的衰减率的猜想。