Generalized Besicovitch Decay for Entropy Solutions of Fractional Degenerate Parabolic–Hyperbolic Equations

We study the large-time behavior of bounded entropy solutions to a class of fractional degenerate parabolic–hyperbolic equations involving ${\Lambda }^{\alpha }={(-\Delta )}^{\alpha /2}$, $0<\alpha <2$, with initial data taken in generalized Besicovitch spaces associated with algebras possessing a mean value. The diffusion is governed by a nonlocal fractional operator and may vanish on nontrivial ranges of the solution variable, so that the equation combines hyperbolic transport with degenerate nonlocal dissipation. Under a suitable non-degeneracy condition involving the flux and the symbol of the fractional diffusion operator, we prove that the entropy solution converges, as time tends to infinity, to the mean value of the initial data in the generalized Besicovitch sense. The analysis is carried out in the framework of ergodic algebras and uses the formulation of generalized Besicovitch spaces on the corresponding compact space. In this realization, the Besicovitch mean is represented by integration on the compact space, and together with the $L^1$-mean contraction principle, it provides the main mechanism for controlling the mean distance along the evolution. After establishing the entropy formulation and the associated $L^1$-mean contraction principle, we derive a fractional kinetic representation adapted to the nonlocal diffusion and analyze a rescaled family of solutions by means of compactness tools suited to the fractional setting. This yields decay in time averages, which is then improved to full asymptotic convergence by the monotonicity of the $L^1$-mean distance to the equilibrium state.