SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2275-2312, January 2020.
We study the asymptotic behavior of solutions of a class of linear nonlocal measure valued differential equations with time delay. Our main result states that the solutions asymptotically exhibit a parabolic-like behavior in the large time, that is precisely expressed in terms of the heat kernel. Our proof relies on the study of a---self-similar---rescaled family of solutions. We first identify the asymptotic behavior of the solutions by deriving a convergence result in the sense of the Young measures. Then we strengthen this convergence by deriving suitable fractional Sobolev compactness estimates. As a by-product, our main result allows us to obtain asymptotic results for a class of piecewise constant stochastic processes with memory.

中文翻译：

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具有时滞的非局部扩散方程的自相似行为

SIAM数学分析杂志，第52卷，第3期，第2275-2312页，2020年1月。
我们研究了一类具有时滞的线性非局部度量值微分方程解的渐近行为。我们的主要结果表明，该解在很大的时间内渐近地表现出抛物线状的行为，这可以用热核来精确表示。我们的证明依赖于对-自相似-缩放后的解决方案系列的研究。我们首先通过得出Young度量意义上的收敛结果来确定解的渐近行为。然后，我们通过推导适当的分数Sobolev紧密度估计来加强这种收敛。作为副产品，我们的主要结果使我们能够获得一类具有记忆的分段恒定随机过程的渐近结果。