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Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type

M. O. Korpusov

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: We consider Cauchy problems for a class of non-linear equations of Sobolev type. It is shown that for such problems there is a critical exponent (depending on the dimension of the space $\mathbb{R}^N$) according to which a weak local solution either exists uniquely or does not exist.

Keywords: Fujita's critical exponent, non-linear equations of Sobolev type, blow-up, local solubility, non-linear capacity.

UDC: 517.538

MSC: 35B44, 35A01, 35G31

Received: 14.08.2014

DOI: 10.4213/im8285

 English version: Izvestiya: Mathematics, 2015, 79:5, 955–1012

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