In this paper, an abstract degenerate hyperbolic equation is considered that includes the semilinear Blackstock–Crighton–Westervelt equation. By proposing a new approach based on strongly continuous semigroups and resolvent families of operators, we prove an explicit representation of the strong and mild solutions for the linearized model by means of a kind of variation of parameters formula. Moreover, we show that under nonlocal initial conditions, the existence of a mild solution of the semilinear equation can be established.

中文翻译：

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抽象的Blackstock–Crighton–Westervelt方程的适定性

在本文中，考虑了一个抽象的简并双曲方程，其中包括半线性Blackstock–Crighton–Westervelt方程。通过提出一种基于强连续半群和算子可分解族的新方法，我们通过一种参数公式的变化，证明了线性化模型的强弱解的显式表示。此外，我们表明在非局部初始条件下，可以建立半线性方程的温和解的存在。