Abstract:
Abstract: This paper develops a comprehensive theory for variable-exponent Bochner spaces Lp(·)([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 (−Δ)s(t) and Levy-type processes
with variable order α(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,ω), yielding compact embedding results in Lp(·,ω)(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.
