Abstract

In this paper, we study an analogue of the Gellerstedt problem for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain. The main method to study this Gellerstedt problem is the Fourier transform. Based on the Fourier transform, we reduce the considering problem to a planar analogue of the Gellerstedt spectral problem with a spectral parameter. The uniqueness of the solution of this problem is proved by the new extreme principle for loaded third-order equations of the mixed type. The existence of a regular solution of the Gellerstedt spectral problem is proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Gellerstedt spectral problem is studied for large values of the spectral parameter. Sufficient conditions are found such that all differentiation operations are legal in this work.

中文翻译：

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无限三维域中三阶抛物线-双曲型方程组的边值问题

摘要

在本文中，我们研究了无限三维域中三阶加载的抛物线-双曲型方程的Gellerstedt问题的类似物。研究此Gellerstedt问题的主要方法是傅里叶变换。基于傅立叶变换，我们将考虑问题简化为带有频谱参数的Gellerstedt频谱问题的平面模拟。新的极限原理针对混合型加载的三阶方程，证明了该问题解决方案的独特性。通过积分方程的方法证明了Gellerstedt谱问题正则解的存在。此外，对于较大的光谱参数值，研究了Gellerstedt光谱问题解的渐近行为。