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In this paper, we prove sharp blow-up and global existence results for a timefractional diffusion-wave equation with a nonlinear memory term in a boundeddomain, where the fractional derivative in time is taken in the sense of Caputotype. Moreover, we also give a result for nonexistence of global solutions to awave equation with a nonlinear memory term in a bounded domain. The proof ofblow-up results is based on the eigenfunction method and the asymptoticproperties of solutions for an ordinary fractional differential inequality.

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain