Loss of regularity for second order hyperbolic equations with singular coefficients. (English) Zbl 1098.35115

The author is interested in the backward Cauchy problem \[ u_{tt}- \lambda(t)^2 b(t)^2\Delta u= 0,\quad u(T,x)= u_0(x),\quad u_t(T, x)= u_1(x) \] in the strip \([0,T]\times\mathbb{R}^n\). The function \(\lambda= \lambda(t)\) describes the decreasing behaviour with a degeneracy at \(t= 0\), the nonnegative function \(b= b(t)\) describes the oscillating behaviour of the coefficient.
The author explains the relation between the interplay of both parts and the loss of regularity of the solution. In the case \(0\leq b_0\leq b(t)\) the quantities (for a suitable \(\kappa_1\geq 0\)) \[ \sup_{t\in (0,T]}\,\Biggl({\Lambda(t)\over \lambda(t)(\text{ln\,} \Lambda(t)^{-1})^{\kappa_1}}|b'(t)|\Biggr),\;\sup_{t\in (0,T]}\,\Biggl(\Biggl({\Lambda(t)\over \lambda(t)(\text{ln\,}\Lambda(t)^{-1})^{\kappa_1}}\Biggr)^2|b''(t)|\Biggr) \] have an essential influence on the loss of regularity.
Finally, the case that \(b= b(t)\) has a countable number of zeros is discussed. The optimality of the results is shown by using Floquet theory.


35L80 Degenerate hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations