Counter-examples of regularity behavior for \(\sigma \)-evolution equations (Q549747)

The authors study the family of Cauchy problems \[ u_{tt}+ a(t)(-\Delta)^\sigma u= 0,\quad u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x),\quad \sigma\in (0,1]. \] Properties of the coefficient \(a\), in particular, oscillations in the time-dependent coefficient, have an important influence on the Sobolev well-posedness of the Cauchy problem. The authors control the first two derivatives of \(a\) and derive a well-posedness result with an at most loss of Sobolev regularity of the solution. An interesting question is to prove the optimality of the loss, that is, that the announced loss really appears. By using Floquet theory the authors construct an example for a coefficient \(a\) which leads to an infinite loss of regularity. So, \(H^\infty\) well-posedness cannot be expected. By using an instability argument the authors construct a sequence of Cauchy problems which explains that a \(\mu\)-loss of regularity really appears. In this way one can understand, for example, that an arbitrary small loss may really appear.