LAUSR

The goal of this paper is to establish a global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients in $$C^2((0,T];C^\infty ({{\mathbb {R}}}^n))$$ C 2 ( ( 0 , T ] ; C ∞ ( R n ) ) growing polynomially in x and singular in t . The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behavior is captured by the blow-up of the first and second t -derivatives of the coefficients which allows the coefficients to be either logarithmic-type or oscillatory-type near $$t=0$$ t = 0 . To arrive at an energy estimate, we perform a conjugation by a pseudodifferential operator of the form $$e^{\nu (t)\Theta (x,D_x)},$$ e ν ( t ) Θ ( x , D x ) , where $$\Theta (x,D_x)$$ 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 $$\nu (t)$$ ν ( t ) gives a scale for the loss. We call the conjugating operator as loss operator and depending on its order we report that the solution experiences zero, arbitrarily small, finite or infinite loss in relation to the initial datum. We also provide a counterexample and derive the anisotropic cone conditions in our setting.