This article is cited in 1 paper

Instantaneous blow-up versus local solubility of the Cauchy problem for a two-dimensional equation of a semiconductor with heating

M. O. Korpusov^ab, A. A. Panin<sup>a

a</sup> Faculty of Physics, Lomonosov Moscow State University
^b Peoples' Friendship University of Russia, Moscow

Abstract: We consider the Cauchy problem for a model third-order partial differential equation with non-linearity of the form $|\nabla u|^q$. We prove that for $q\in(1,2]$ the Cauchy problem in $\mathbb{R}^2$ has no local-in-time weak solution for a large class of initial functions, while for $q>2$ a local weak solution exists.

Keywords: finite-time blow-up, non-linear waves, instantaneous blow-up.

UDC: 517.957

MSC: Primary 35B44; Secondary 35K30, 35D30

Received: 12.10.2018

DOI: 10.4213/im8872

 English version: Izvestiya: Mathematics, 2019, 83:6, 1174–1200

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