Nonexistence results for a pseudo-hyperbolic equation in the Heisenberg group

Abstract

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear pseudo-hyperbolic equation $$ u_{tt} -\Delta_{\mathbb H} u_{tt}-\Delta_{\mathbb H} u=|u|^p, \quad (\eta, t) \in \mathbb{H} \times (0,\infty), \; p>1, $$ where $\Delta_\mathbb{H}$ is the Kohn-Laplace operator on the $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$. Then, this result is extended to the case of a $2 \times 2$-system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions.

Keywords

• Nonexistence
• nonlinear pseudo-hyperbolic equation
• systems of pseudo-hyperbolic equations
• Heisenberg group