We use a generalized Riemann-Liouville type derivative of an abstract function of one variable and existence of a weak solution to an abstract fractional parabolic problem on [0, T] containing Riemann-Liouville derivative of a function of one variable and spectral fractional powers of a weak Dirichlet-Laplace operator to study existence of a strong solution to this problem. Our goal in this regard is to provide conditions that allow the transition from a weak to a strong solution. Next, we passage from the abstract problem to a classical one on \([0,T]\times \varOmega \), containing partial (with respect to time \(t\in [0,T]\,\)) Riemann-Liouville derivative of the unknown real-valued function of two variables and fractional powers of a weak Dirichlet-Laplacian of this function (with respect to spatial variable \(x\in \varOmega \)). The most important in this regard is a theorem on the relation of the fractional derivatives of an abstract function of one variable and real-valued one of two variables.

我们使用一个变量的抽象函数的广义 Riemann-Liouville 型导数，并且存在一个 [0， T] 上的抽象分数抛物线问题的弱解，其中包含一个变量的函数的 Riemann-Liouville 导数和一个弱狄利克雷-拉普拉斯算子的谱分数幂来研究这个问题的强解的存在。我们在这方面的目标是提供条件，允许从弱解决方案过渡到强解决方案。接下来，我们从抽象问题过渡到 \（[0，T]\times \varOmega \） 的经典问题，其中包含两个变量的未知实值函数的部分（相对于时间 \（t\in [0，T]\，\））黎曼-刘维尔导数和该函数的弱狄利克雷-拉普拉斯算子的分数幂（相对于空间变量 \（x\in \varOmega \））).在这方面，最重要的是关于一个变量的抽象函数和两个变量中的一个实值函数的分数导数关系的定理。