Fractional elliptic and parabolic equations-Liouville theorems, regularities, and a priori estimates

In this paper, we present a summary of our recent advancements in the study of various fractional elliptic and parabolic equations, with a focus on Liouville type theorems, integral representations of solutions, interior regularities, blow-up and re-scaling analysis, and a priori estimates.These results and their proofs have been previously published in various sources. Here, we adopt an intuitive approach, using clear language and visual illustrations to highlight the inner connections among these results and the ideas behind their proofs. Our goal is to provide readers with a comprehensive understanding of these approaches, enabling them to effectively apply the useful methods to solve a wide range of related problems in this field.