Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces

Let G be a graded Lie group with homogeneous dimension Q. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator R of homogeneous degree ν≥ 2 on G with power type nonlinearity |u|ᵖ and initial data taken from negative order homogeneous Sobolev space H-γ( G), γ>0. In the framework of Sobolev spaces of negative order, we prove that pCrit(Q, γ, ν) :=1+2ν/Q+2γ is the new critical exponent for γ∈ (0, Q/2). More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for p>pCrit(Q, γ, ν) in the energy evolution space C([0, T], Hˢ(G)), s∈ (0, 1]. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for 1<p<pCrit(Q, γ, ν). Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on Rⁿ, and more generally, on stratified Lie groups.