Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups
Copyright policy )Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups
In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on $\mathbb R^n$ and on the Heisenberg group, such as powers of the Laplacian or of the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if $G$ is a graded Lie group of homogeneous dimension $Q$ and $a>0$, $1
21 pages
- Ghent University Belgium
- Imperial College London United Kingdom
- Institute of Mathematics and Mathematical Modeling Kazakhstan
- Al-Farabi Kazakh National University Kazakhstan
- Queen Mary University of London United Kingdom
Higher-order nonlinear hyperbolic equations, Rockland operator, General Mathematics, 35L75, SPACES, math.FA, 35L71, Heisenberg group, 510, 0101 Pure Mathematics, Mathematics - Analysis of PDEs, Second-order semilinear hyperbolic equations, 35L71, 35L75, 35R03, 22E25, 0102 Applied Mathematics, FOS: Mathematics, 22E25, 35R03, Gagliardo-Nirenberg inequality, math.AP, Sub-Laplacian, graded group, Science & Technology, Hyperbolic equations on manifolds, HARDY, PDEs on Heisenberg groups, Lie groups, Carnot groups, etc., Initial value problems for second-order hyperbolic equations, Nonlinear damped wave equation, Functional Analysis (math.FA), Mathematics - Functional Analysis, Analysis on real and complex Lie groups, Mathematics and Statistics, Graded group, Physical Sciences, INEQUALITIES, Mathematics, Analysis of PDEs (math.AP)
Higher-order nonlinear hyperbolic equations, Rockland operator, General Mathematics, 35L75, SPACES, math.FA, 35L71, Heisenberg group, 510, 0101 Pure Mathematics, Mathematics - Analysis of PDEs, Second-order semilinear hyperbolic equations, 35L71, 35L75, 35R03, 22E25, 0102 Applied Mathematics, FOS: Mathematics, 22E25, 35R03, Gagliardo-Nirenberg inequality, math.AP, Sub-Laplacian, graded group, Science & Technology, Hyperbolic equations on manifolds, HARDY, PDEs on Heisenberg groups, Lie groups, Carnot groups, etc., Initial value problems for second-order hyperbolic equations, Nonlinear damped wave equation, Functional Analysis (math.FA), Mathematics - Functional Analysis, Analysis on real and complex Lie groups, Mathematics and Statistics, Graded group, Physical Sciences, INEQUALITIES, Mathematics, Analysis of PDEs (math.AP)
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