The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (156 KB) | Abstract as PDF | Metadata | Table of Contents | ZAA summary
Volume 25, Issue 2, 2006, pp. 131–142
DOI: 10.4171/ZAA/1281

Published online: 2006-06-30

Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping

Mokhtar Kirane[1] and Nasser-edine Tatar[2]

(1) Université de la Rochelle, France
(2) King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in $Q:=(0,\infty )\times \mathbb{R}^{N},$ where $D_{+}^{\alpha }u$, $% 0<\alpha <1$ is a time fractional derivative, with given initial position and velocity $u(0,x)=u_{0}(x)$ and $u_{t}(0,x)=u_{1}(x).$ We find the Fujita's exponent which separates in terms of $p,\alpha $ and $N,$ the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on $u_{1}(x)$ and $h(x,t)$ assuring non-existence of local solutions.

Keywords: Fractional damping, non-existence, nonlinear hyperbolic equations

Kirane Mokhtar, Tatar Nasser-edine: Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping. Z. Anal. Anwend. 25 (2006), 131-142. doi: 10.4171/ZAA/1281