In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

在这项研究中，我们研究了具有非线性源项的时间分数四阶方程的初值问题 (IVP)。更具体地说，我们考虑具有指数非线性的时间分数双谐波和时间分数 Cahn-Hilliard 方程。利用傅里叶变换的概念，证明了温和解的广义公式以及分解算子的平滑效果。对于与第一个关联的 IVP，通过使用带有函数$\Xi (z)={\textrm {e}}^{|z|^{p}}-1$的 Orlicz 空间以及它与通常的 Lebesgue 空间之间的一些嵌入，我们证明该解决方案是一个全局时间解决方案，或者如果初始值是规则的，它将在有限时间内爆炸。在奇异初始数据的情况下，导出了本地时间/全局时间的存在性和唯一性。此外，研究了温和溶液的规律性。对于与第二个相关的IVP，对广义公式进行了一些修改以处理非线性项。我们还为分解算子的导数建立了一些重要的估计，它们是使用 Picard 序列证明解的局部时间存在性的基础。