Instantaneous Blow-Up of a Weak Solution of a Problem in Plasma Theory on the Half-Line

Abstract

We consider a problem with some boundary and initial conditions for an equation arising in the theory of ion-sound waves in plasma. We prove that if the spatial (one-dimensional) variable ranges on an interval, then this problem has a unique nonextendable classical solution which in general exists only locally in time. If the spatial variable varies on the half-line, then, for the problem in question, we obtain an upper bound for the lifespan of its weak solution and find initial conditions for which there exist no solutions even locally in time (instantaneous blow-up of the weak solution). A similar result is obtained for the classical solution.