The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization of singularities to obtain an asymptotic Erdélyi series in inverse powers of the cubic root of the time variable with coefficients depending on the self-similar variable and the logarithm of time.

中文翻译：

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演化Airy方程Cauchy问题解的渐近性。

建立了Airy方程（一个三阶演化方程）的Cauchy问题的解在大部分时间的渐近行为。我们假设初始函数是局部Lebesgue可积的，并且在无穷大处具有幂律渐近性。对于与Airy函数进行卷积积分形式的解，我们使用辅助参数方法和奇异性正则化来获得时间变量的三次根的反幂的渐近Erdélyi级数，其系数取决于自变量。类似的变量和时间的对数。