In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted $L^p$-path spaces is proved. In particular, as special cases the classical Caputo derivative and other fractional derivatives appearing in applications are included. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type$\frac{d}{dt}(k⁎u)(t)+A(t,u(t))=f(t),0<t<T,$ with (in general nonlinear) operators $A(t,\cdot )$ satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on $[0,\infty )$ with $\underset{s\to \infty }{\lim }k(s)=0$. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators and the time-fractional (stochastic) p-Laplace equation are covered.

在本文中，时间加权的 Gelfand 三元组上广义时间分数导数的强耗散性 ${升}^{磷}$-path 空间被证明。特别是，作为特殊情况，经典的 Caputo 导数和应用中出现的其他分数导数都包括在内。因此，我们获得了具有广义时间分数阶导数的 Gelfand 三元组上演化方程解的存在性和唯一性。这些方程是类型$\frac{d}{d吨}(克⁎你)(吨)+\mathrm{一种}(吨,你(吨))=F(吨),0<吨<吨,$ 使用（一般为非线性）运算符 $\mathrm{一种}(吨,\cdot )$满足一般弱单调性条件。这里k是一个非递增的局部 Lebesgue 可积非负函数$[0,\infty )$ 和 $\underset{秒\to \infty }{\mathrm{林}}克(秒)=0$. 对于f被时间分数加性噪声代替的情况，也获得了类似的结果。应用包括广义时间分数准线性（随机）偏微分方程。特别是，涵盖了具有普通或分数拉普拉斯算子的时间分数（随机）多孔介质和快速扩散方程以及时间分数（随机）p-拉普拉斯方程。