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On a differential equation involving Hilfer-Hadamard fractional derivative. (English) Zbl 1255.34007
This 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)].

MSC:
34A08 Fractional ordinary differential equations
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