Global Well-Posedness and Regularity Issues Associated with Singular Hyperbolic Cauchy Problems

In this talk, we consider Cauchy problems for a class of singular hyperbolic equations whose coefficients or their t-derivatives tend to infinity in some sense as t → 0 or |x| → \infty (here the variables t and x represent time and space, respectively, and (t,x) \in [0,T] \times \mathbb{R}^n). In particular, our interest is either blow-up or infinitely many oscillations near t = 0 and polynomial growth in x variable of the coefficients. In order to study the interplay of the singularity in t and unboundedness in x, we consider a special class of metrics on the phase space and build a parameter dependent pseudodifferential calculus related to the metric. We derive energy estimates to establish global well-posedness and loss of regularity for the Cauchy problems in the appropriate Sobolev spaces. Our methodology relies upon two important techniques: the subdivision of the extended phase space using the Planck function associated to the metric and conjugation of a first order system corresponding to the singular equation. In order to overcome the difficulty of tracking a precise loss in our context we introduce a class of parameter dependent pseudodifferential operators of the form e^{ν(t) Θ(x,D_x)} for the purpose of conjugation. This operator compensates, microlocally, the loss of regularity of the solutions. The operator Θ(x,D_x) explains the quantity of the loss by linking it to the metric on the phase space and the singular behavior, while ν(t) gives a scale for the loss. We establish that the metric governing the conjugated operator is conformally equivalent to the initial metric where the conformal factor is given by the symbol of the operator Θ(x,D_x). Depending on the order of the conjugating operator, we report that the solution experiences zero, arbitrarily small, finite or infinite loss of regularity in relation to the initial datum.