Geodesic
On the critical exponent ``instantaneous~blow-up'' versus ``local solubility'' in the Cauchy~problem for a~model equation of Sobolev type
Izvestiya. Mathematics , Tome 85 (2021) no. 1, pp. 111-144.

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We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form $|\nabla u|^q$. We prove that when $q\in(1,3/2]$ the Cauchy problem in $\mathbb{R}^3$ has no local-in-time weak solution for a large class of initial functions, while when $q>3/2$ there is a local weak solution.
Keywords: finite-time blow-up, non-linear waves, instantaneous blow-up. 
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M. O. Korpusov; A. A. Panin; A. E. Shishkov. On the critical exponent ``instantaneous~blow-up'' versus ``local solubility'' in the Cauchy~problem for a~model equation of Sobolev type. Izvestiya. Mathematics , Tome 85 (2021) no. 1, pp. 111-144. http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a4/

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