Electron. J. Diff. Equ., Vol. 2015 (2015), No. 227, pp. 1-10.

Nonexistence of global solutions of some nonlinear space-nonlocal evolution equations on the Heisenberg group

Bashir Ahmad, Ahmed Alsaedi, Mokhtar Kirane

Abstract:
This article presents necessary conditions for the existence of weak solutions of the following space-nonlocal evolution equations on $\mathbb{H}\times(0, +\infty)$, where $\mathbb{H}$ is the Heisenberg group:
$$\displaylines{ \frac{\partial^2 u }{\partial t^2} + (- \Delta_{\mathbb{H}})^{\alpha/2}|u|^m = |u|^{p},\cr \frac{\partial u}{\partial t} + (- \Delta_{\mathbb{H}})^{\alpha/2} |u|^m = |u|^{p},\cr \frac{\partial^2 u }{\partial t^2} + (- \Delta_{\mathbb{H}})^{\alpha/2} |u|^m + \frac{\partial u }{\partial t} = |u|^p, }$$
$p \in \mathbb{R}, p>1, m \in \mathbb{N}$. Moreover, the life span for each equation is estimated under some suitable conditions. Our method of proof is based on the test function method.

Submitted August 19, 2015. Published September 2, 2015.
Math Subject Classifications: 35A01, 35R03, 35R11
Key Words: Nonlinear hyperbolic equations; nonlinear parabolic equations; nonlocal operators; damping; blowing-up solutions.

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Bashir Ahmad
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group
Department of Mathematics, Faculty of Science
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: bashirahmad_qau@yahoo.com
  Ahmed Alsaedi
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group
Department of Mathematics, Faculty of Science
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
email: aalsaedi@hotmail.com
Mokhtar Kirane
Laboratoire de Mathématiques, Image et Applications
Université de La Rochelle, Avenue M. Crépeau
17042 La Rochelle Cedex, France
email: mokhtar.kirane@univ-lr.fr

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