Abstract
We consider a fractional-in-time evolution equation arising in the theory of ion-sound waves in plasma, where the spatial variable varies on the half-line. We first provide sufficient conditions for which there exist no global-in-time weak solutions and obtain an upper bound of the lifespan. Next, we find a class of initial data for which local-in-time weak solutions do not exist, that is, an instantaneous blow-up of weak solutions occurs. The proofs of our results are based on some integral inequalities and the test function method.