In this talk, I will discuss the semilinear hypoelliptic damped wave equation with power-type nonlinearity associated with a Rockland operator on graded Lie groups. Specifically, we will concentrate on the case when the initial data belongs to Sobolev spaces of negative order. We show the global-in-time existence of small data Sobolev solutions of lower regularity for the supercritical range and a finite-time blow-up of weak solutions for the subcritical range. For the particular settings of the Heisenberg group and Euclidean space, we will prove that the critical exponent belongs to the blow-up case. Furthermore, to precisely characterize the blow-up time, we derive sharp upper and lower bound estimates for the lifespan in the subcritical cases.

This talk is based on my joint research with Aparajita Dasgupta (IIT Delhi), Shyam Swarup Mondal (ISI Kolkata), Michael Ruzhansky (Ghent University), and Berikbol Torebek (Ghent University).

The video of this talk is available on the IISc Math Department channel.