Compactness and maximal regularity in variable-exponent bochner spaces with applications to nonlocal evolution equations

This paper develops a comprehensive theory for variable-exponent Bochner spaces\ (L^p () (0, T;X) \), establishing fundamental results on compact embeddings and maximal regularity with applications to nonlocal evolution equations. We extend the classical Aubin-Lions framework through innovative modular convergence techniques, proving sharp compactness criteria under log-Holder continuity conditions. For time-dependent fractional operators, including the fractional Laplacian\ ( (-) ^s (t) \) and Levy-type processes with variable order\ ( (t) \), we derive optimal maximal regularity estimates that reveal new connections between exponent functions\ (p (t) \) and operator orders. A groundbreaking contribution is our systematic analysis of fractal dimension dynamics in variable-order fractional PDEs, characterizing how evolving regularity\ (s (t) \) governs solution behavior. Furthermore, we develop novel functional-analytic tools for stochastic exponents\ (p (t, ) \), yielding compact embedding results in\ (L^p (, ) (X) \) spaces and boundedness properties for nonlinear operators. Combining techniques from modular function theory, refined interpolation methods, and stochastic analysis, our work provides powerful new approaches for problems in anomalous diffusion and heterogeneous media. These results significantly advance both the theoretical foundations and practical applications of variable-exponent spaces in modern PDE analysis.