On a differential equation involving Hilfer-Hadamard fractional derivative.

*(English)*Zbl 1255.34007This paper deals with a non-existence result for global solutions to the problem
\[
D^{\alpha,\beta}_{a}u(t)=f(t,u(t)),\;D^{\beta-1,1-\alpha}_{a}u(a)=u_0\geq 0,
\]
where \(0<\alpha<1\), \(0\leq \beta\leq 1\), \(t>a>0\) and \(u_0\) is a given function. The fractional derivative \(D^{\alpha,\beta}_{a}\) is new and defined as an interpolation of the Hadamard fractional derivative and its Caputo counterpart.

Actually, the result is obtained for a function \(f\) which satisfies \( f(t,u(t))\geq (\log (t/a))^\mu|u(t)|^m\) for some \(m>1\) and \(\mu\in \mathbb{R}\). The proof uses the test function method developed by E. Mitidieri and S. I. Pokhozhaev [A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities (Russian). Moskva: MAIK Nauka/Interperiodica (2001; Zbl 0987.35002)].

Actually, the result is obtained for a function \(f\) which satisfies \( f(t,u(t))\geq (\log (t/a))^\mu|u(t)|^m\) for some \(m>1\) and \(\mu\in \mathbb{R}\). The proof uses the test function method developed by E. Mitidieri and S. I. Pokhozhaev [A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities (Russian). Moskva: MAIK Nauka/Interperiodica (2001; Zbl 0987.35002)].

Reviewer: Gisèle M. Mophou (Pointe-à-Pitre)

##### MSC:

34A08 | Fractional ordinary differential equations |

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\textit{M. D. Qassim} et al., Abstr. Appl. Anal. 2012, Article ID 391062, 17 p. (2012; Zbl 1255.34007)

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