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 Ḣ^-γ(𝔾), γ>0. In the framework of Sobolev spaces of negative order, we prove that p_Crit(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>p_Crit(Q, γ, ν) in the energy evolution space 𝒞([0, T], H^s(𝔾)), s∈ (0, 1]. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for 1更多