Counterexample of loss of regularity for fractional order evolution equations with both degenerating and oscillating coefficients

For weak evolution models of fractional order with singularity near the origin, the joint influence from the principal σ-Laplacian operator, degenerating part and oscillating part is of prime concern in the discussion of regularity behavior of the solutions. We apply the techniques from the micro-local analysis to explore the upper bound of loss of regularity. Furthermore, in order to demonstrate the optimality of the estimates, a delicate counterexample with periodic coefficients will be constructed to show the lower bound of loss of regularity by the application of harmonic analysis and instability arguments. This optimality discussion develops the theory in Cicognani and Colombini (2006), Cicognani et al. (2008), Lu and Reissig (2009) and Lu and Reissig (2009) by combining both oscillation and degeneracy of the coefficients