On the blowing up solutions of a system of fractional differential equations
Sofwah Ahmad and and Mokhtar Kirane

Department of Mathematics, College of Computing and Mathematical Sciences, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, UAE

Abstract

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You Can Edit It Again Later In this paper, we are concerned with the study of the system of fractional differential equations
\begin{align*}
^cD_{0+}^{\alpha} X(t) & =\Gamma(\alpha)(t+1)^{k_1}X^{a}(t)Y^{q}(t),
%%
\quad 0<\alpha<1, \quad t>0, \\
%%
^cD_{0+}^{\beta} Y(t) & =\Gamma(\beta)(t+1)^{k_2}Y^{b}(t)X^{p}(t),
\quad 0<\beta<1, \quad t>0, \\
\label{eq:4-frac sys_sigma}
\end{align*}
subject to
\begin{equation*}
X(0) =X_{0}>0,\quad Y(0)=Y_{0}>0,
\label{eq3:initial_value}
\end{equation*}
where \(\Gamma(\sigma)\) stands for the Gamma function, \(^cD_{0+}^{\alpha}\) stands for the Caputo fractional derivative, and \(a,b,p,q, k_1,\) and \(k_2\) are real numbers that will be specified later. We present sufficient conditions for the non-existence of global solutions. Furthermore, we present the asymptotic growth of blowing-up solutions near the blow up time. More precisely, we give the growth rate estimate near the blow-up time by employing Parseval^s formula for the Mellin transform.

Keywords: Fractional derivative, Non-existence of global solutions, Asymptotic growth of blowing up solutions, Blow-up time.

Topic: Minisymposia Differential Equations