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

2024/04/12 by Dasgupta, Aparajita, Kumar, Vishvesh, Mondal, Shyam Swarup +1
#35A01 #35B33 #35B44 #35L15 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 43A80 #Secondary 35L15

paper · doi:10.48550/arxiv.2404.08766

Abstract

Let \mathbb 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 \mathbb G with power type nonlinearity |u|^p and initial data taken from negative order homogeneous Sobolev space H^-γ(\mathbb 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(\mathbbG)), s∈ (0, 1]. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for 1

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