## 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.

### MSC:

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