具有含时生成元的偏微分方程组温和解的存在性

我们给出了弱耦合系统正温和解全局存在的充分条件：\begin{eqnarray*} \frac{\partial u_{1}}{\partial t} &=&\rho_{1}t^{\rho_{1}-1}\Delta_{\alpha_{1}}u_{1}+t^{\sigma_{1}}u_{2}^{\beta_{1}},\ \ u_{1}\left(0\right) =\varphi_{1}, \\ \frac{\partial u_{2}}{\partial t} &=&\rho_{2}t^{\rho_{2}-1}\Delta_{\alpha_{2}}u_{2}+t^{\sigma_{2}}u_{1}^{\beta_{2}},\ \ u_{2}\left(0\right) =\varphi_{2}, \end{eqnarray*} 其中$\Delta_{\alpha_{i}}$是分数阶 Laplacian，$0<\alpha_{i}\leq 2,\ \beta_{i}>1,\ \rho_{i}>0,\ \sigma_{i}>-1\$为常数，且初始数据$\varphi_{i}$为正、有界且可积的函数。