Global Well-Posedness with Loss of Regularity for a Class of Singular Hyperbolic Cauchy Problems

The goal of this paper is to summarize global well-posedness results for a class of strictly hyperbolic Cauchy problems with coefficients in \(L^1\left ([0, T] ; C^{\infty }\left (\mathbb {R}^n\right )\right )\) growing polynomially in x and singular in t. The problems we study are of the strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behaviour is captured by the blow-up of the first and second t-derivatives of the coefficients, which allows the coefficients to either blow up or oscillate near \(t=0\) . A crucial step in arriving at the results is conjugation by a pseudodifferential operator of the form \(e^{\nu (t) \Theta \left (x, D_x\right )}\) , where \(\Theta \left (x, D_x\right )\) explains the quantity of the loss by linking it to the metric on the phase space and the singular behaviour, while \(v(t)\) gives a scale for the loss. Depending on the order of this operator, the solution experiences zero, arbitrarily small, finite or infinite loss in relation to the initial datum. We also show the anisotropic cone conditions in our setting.