In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

在希尔伯特空间中，我们考虑一类具有时间变量t的 Caputo 分数导数的非线性分数方程自伴正无界算子的空间分数函数。我们考虑具有时间奇异系数的全局 Lipschitz 和局部 Lipschitz 源的各种情况。这些来源推广了众所周知的分数方程，例如分数 Cahn-Allen 方程、分数 Burger 方程、分数 Cahn-Hilliard 方程、分数 Kuramoto-Sivashinsky 方程等。在适当的假设下，我们研究存在，最大解的唯一性，以及关于扰动分数阶问题的解的稳定性。我们还建立了一些全局存在性并证明了全局解可以通过已知的渐近函数来近似为\(t\rightarrow \infty \)。