Nonexistence of global weak solutions of a system of wave equations on the Heisenberg group

Volume 9, Issue 5, pp 3279--3286 http://dx.doi.org/10.22436/jnsa.009.05.114

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Authors

Rabah Kellil - Faculty of Science Al Zulfi, University of Majmaah, Al Zulfi, KSA. Mokhtar Kirane - Laboratoire de Mathématiques, Image et Applications, Faculté des Sciences, pôle Sciences et Technologies, Avenue M. Crépeau, 17000, La Rochelle, France.

Abstract

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudoparabolic equation \[u_{tt} -\Delta_{\mathbb{H}}u + (- \Delta_{\mathbb{H}})^{ \frac{\delta}{2} }u = f(\eta; t)u^p;\qquad (\eta, t) \in \mathbb{H} \times(0,\infty); p > 1;\quad u \geq 0;\] where \( \Delta_{\mathbb{H}}\) is the Kohn{Laplace operator on the \((2N + 1)\)-dimensional Heisenberg group \(\mathbb{H}\) and \(f(\eta; t)\) is a given function. 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 a duality argument.

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Keywords

• Nonexistence
• nonlinear hyperbolic equation
• systems of hyperbolic equations
• Heisenberg group.

MSC

•  47J35
•  34A34
•  35R03.

References

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