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In this paper, we are concerned with the study of the system of fractional differential equations \(\begin{aligned} ^cD_{0+}^{\alpha } X(t) & =\Gamma (\alpha )(t+1)^{k_1}X^{a}(t)Y^{q}(t), \quad 0<\alpha <1, \quad t>0, \\ ^cD_{0+}^{\beta } Y(t) & =\Gamma (\beta )(t+1)^{k_2}Y^{b}(t)X^{p}(t), \quad 0<\beta <1, \quad t>0, \end{aligned}\) subject to \(\begin{aligned} X(0) =X_{0}>0,\quad Y(0)=Y_{0}>0, \end{aligned}\) 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.

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On the Blowing-Up Solutions of a System of Fractional Differential Equations

  • 会议论文

  • 发表日期：

    2025年

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  • Sofwah Ahmad,
  • Mokhtar Kirane

摘要

In this paper, we are concerned with the study of the system of fractional differential equations \(\begin{aligned} ^cD_{0+}^{\alpha } X(t) & =\Gamma (\alpha )(t+1)^{k_1}X^{a}(t)Y^{q}(t), \quad 0<\alpha <1, \quad t>0, \\ ^cD_{0+}^{\beta } Y(t) & =\Gamma (\beta )(t+1)^{k_2}Y^{b}(t)X^{p}(t), \quad 0<\beta <1, \quad t>0, \end{aligned}\) subject to \(\begin{aligned} X(0) =X_{0}>0,\quad Y(0)=Y_{0}>0, \end{aligned}\) 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.