Counter-examples of regularity behavior for σ-evolution equations

In this paper, we mainly discuss the infinite loss of regularity and μ-loss for a σ-evolution type model with oscillating in time coefficients. On the one hand, an explicit counter-example has been constructed in the frequency space to show the precise infinite loss of regularity. On the other hand, due to the finite propagation speed property for σ∈(0,1], we construct the counter-example of a sequence of solutions in R by applying state of art techniques.