Geodesic
Instantaneous blow-up versus local solubility of~the Cauchy problem for a~two-dimensional equation
Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1174-1200

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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. 
@article{IM2_2019_83_6_a2,
     author = {M. O. Korpusov and A. A. Panin},
     title = {Instantaneous blow-up versus local solubility of~the {Cauchy} problem for a~two-dimensional equation},
     journal = {Izvestiya. Mathematics },
     pages = {1174--1200},
     publisher = {mathdoc},
     volume = {83},
     number = {6},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_6_a2/}
}
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M. O. Korpusov; A. A. Panin. Instantaneous blow-up versus local solubility of~the Cauchy problem for a~two-dimensional equation. Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1174-1200. http://geodesic.mathdoc.fr/item/IM2_2019_83_6_a2/