On the optimal regularity of coefficients in hyperbolic Cauchy problems. (Q1407219)

The authors consider the Cauchy problem for a second-order hyperbolic equation \[ \begin{gathered} u_{tt}(t, x)- a(t) u_{xx}(t, x)+ b(t) u_x(t, x)= 0,\quad [0, T]\times \mathbb{R}^n,\\ u(0, x)= \phi(x),\quad u_t(0, x)= \psi(x).\end{gathered} \] They prove that if the coefficients \(a(t)> 0\) and \(b(t)\) satisfy \[ \begin{alignedat}{2} |\partial^k_t a(t)| &\leq A_k\Biggl({1\over t}\log{1\over t}\Biggr)^k,\quad && k= 0,1,\\ |\partial^k_t b(t)| &\leq B_k\Biggl({1\over t}\log{1\over t}\Biggr)^{k+ 1},\quad && k= 0,1\end{alignedat} \] near \(t= 0\), where \(A_k\), \(B_k\) are nonnegative constants, then the above Cauchy problem is \(C^\infty\)-well-posed. Moreover, they show by examples that the above conditions on \(a\), \(b\) are nearly necessary conditions for the Cauchy problem to be \(C^\infty\)-well-posed.