BLOW UP RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS AND SYSTEMS

Ali Hakem and Mohamed Berbiche

Abstract: The aim of this research paper is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential equation \begin{gather*} \mathbf{D}_{0|t}^{\alpha}u+(-\Delta)^{\beta/2}|u|^{m-1}u+a(x)\cdot\nabla|u|^{q-1}u=h(x,t)|u|^p, (t,x)\in Q,
u(0,x)=u_0(x), x\in\R^N \end{gather*} where $(-\Delta)^{\beta/2}$, $0<\beta<2$ is the fractional power of $-\Delta$, and $\mathbf{D}_{0|t}^{\alpha}$, $(0<\alpha<1)$ denotes the time-derivative of arbitrary $\alpha\in(0;1)$ in the sense of Caputo. The results are shown by the use of test function theory and extended to systems of the same type.

Keywords: blow-up; fractional derivatives; critical exponent

Classification (MSC2000): 58J45, 26A33, 35B44

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