Decay estimates for the linear damped wave equation on the Heisenberg group

Abstract

This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group ${\mathbf{H}}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1({\mathbf{H}}_n)$ regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of ${\mathbf{H}}_n$ and on the properties of the Hermite functions that form a maximal orthonormal system for $L^2({\mathbb{R}}^n)$ of eigenfunctions of the harmonic oscillator.

References in corpus (5)

• Wave equation for sums of squares on compact Lie groups
• Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups
• Sharp $L^p$-bounds for the wave equation on groups of Heisenberg type
• Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
• Wave equations on graded groups and hypoelliptic Gevrey spaces

Cited by in corpus (6)

• Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups
• On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order
• A global existence result for a semilinear wave equation with lower order terms on compact Lie groups
• Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group
• Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces