Solutions analytic in time are not dense (Q1327080)

For the wave equation \[ (\partial_ t + a \partial_ x)u = 0, \] with \(a(t,x)\) real valued, we ask whether the solutions which are real analytic with respect to the time variable \(t\) are dense. The answer depends on the regularity of the coefficient \(a\). For example, if \(a\) is real analytic in \(t\), \(x\), the Cauchy-Kovalevsky theorem implies that the solutions real analytic in \(t\), \(x\) are dense. Somewhat surprisingly, the same positive result is valid when \(a = a(x)\) is only \(C^ 1\) in \(x\). We show that the hypothesis that \(a\) is independent of \(t\) cannot be replaced by the weaker assumption that \(a\) is real analytic in \(t\), even when it is \(C^ \infty\) in \(x\).