AI 中文总结
该研究针对带时空非局部算子的分数阶抛物方程，建立加权混合范数估计，推导奇混合范数空间对应结果，证明其鲁棒性与唯一可解性，采用逐方向延拓论证完成证明。
AI 中文摘要
我们为具有时间和空间双非局部算子的分数阶抛物方程建立了加权混合范数估计，方程为∂_t^α u = Lu - λu + f，定义域为(0,T)×ℝ^d。其中∂_t^α是阶数α∈(0,1)的Caputo导数，L是阶数σ∈(0,2)的空间非局部算子，其核仅在时间上可测。我们还得到了奇混合范数空间中的对应估计，该空间内积分沿时间、外积分沿空间进行。当α→1和σ→2时，这些估计具有鲁棒性；当T<∞或λ>0时，我们还建立了方程的唯一可解性。证明基于逐方向延拓论证。
英文摘要
We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^αu=Lu-λu+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, $\partial_t^α$ is the Caputo derivative of order $α\in(0,1)$, and $L$ is a spatially nonlocal operator of order $σ\in(0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $α\to1$ and $σ\to2$. We also establish unique solvability when either $T<\infty$ or $λ>0$. The proof is based on a direction-by-direction extension argument.
Comments46 pages
