On an instantaneous blow-up of solutions of evolutionary problems on the half-line

M. O. Korpusov

Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 914-930

Résumé

We consider some initial-boundary value problems on the half-line for ‘1+1’-dimensional equations of Sobolev type with homogeneous boundary conditions at the beginning of the half-line. We show that weak solutions of these problems are absent even locally in time. Moreover, we consider problems on an interval with the same boundary conditions on one of the ends of the interval $[0,L]$. We prove the local in time (unique) solubility of the problems under consideration in the classical sense, and obtain sufficient conditions for the blow-up of these solutions in finite time. Using the upper bounds thus obtained for the blow-up times for classical solutions of the corresponding problems, we show that the blow-up time tends to zero as $L\to+\infty$. Thus, a classical solution on the line is also absent, even locally, and we describe an algorithm for the subsequent numerical diagnosis of the instantaneous blow-up on the half-line.

Détail

Keywords: non-linear equations of Sobolev type, blow-up, local solubility, non-linear capacity, bounds for the blow-up time.