Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

Authors

Aparajita Dasgupta, Indian Institute of Technology Delhi
Vishvesh Kumar, Universiteit Gent
Shyam Swarup Mondal, Universiteit Gent
Michael Ruzhansky, Universiteit Gent

Article Type

Research Article

Publication Title

Journal of Evolution Equations

Abstract

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian L on the Heisenberg group Hⁿ with power type nonlinearity |u|^p and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space H˙L^-γ(Hⁿ),γ>0, on Hⁿ. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent pcrit(Q,γ)=1+4Q+2γ, for γ∈(0,Q2), where Q:=2n+2 is the homogeneous dimension of Hⁿ. More precisely, we establish A global-in-time existence of small data Sobolev solutions of lower regularity for p>pcrit(Q,γ) in the energy evolution space; A finite time blow-up of weak solutions for 1crit(Q,γ) under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

DOI

10.1007/s00028-024-00976-5

Publication Date

9-1-2024

Comments

Open Access; Green Open Access; Hybrid Gold Open Access

Recommended Citation

Dasgupta, Aparajita; Kumar, Vishvesh; Mondal, Shyam Swarup; and Ruzhansky, Michael, "Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order" (2024). Journal Articles. 5074.
https://digitalcommons.isical.ac.in/journal-articles/5074