In this paper, we study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations, which the fractional derivatives in time are taken in the sense of Caputo type. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solution. We also establish a well-posed result for nonlinear problem. By applying finite dimensional approximation method, a compact result of solution operators is presented, following this, an existence criterion shows that the Lipschitz condition or smoothness of nonlinear force functions in some literatures can be removed. As an application, we discuss a case of time fractional telegraph equations.

中文翻译：

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分数阶阻尼波动方程的适定性和正则性

在本文中，我们研究了一类时间分数阶阻尼波动方程的温和解的适定性和正则性，其中时间分数阶导数是在Caputo类型的意义上取的。引入温和解的概念来证明线性问题的存在性以及解的规律性。我们还为非线性问题建立了适定结果。通过应用有限维近似方法，给出了解算子的紧凑结果，随后，存在性准则表明可以去除某些文献中的Lipschitz条件或非线性力函数的平滑性。作为一个应用，我们讨论时间分数电报方程的一个例子。