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Instantaneous blow-up of a weak solution of a problem in plasma theory on the half-line. (English. Russian original) Zbl 1416.35046

Differ. Equ. 55, No. 1, 60-67 (2019); translation from Differ. Uravn. 55, No. 1, 59-66 (2019).
Summary: 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.

MSC:

35B44 Blow-up in context of PDEs
82D10 Statistical mechanics of plasmas
35G31 Initial-boundary value problems for nonlinear higher-order PDEs
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References:

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