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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_2021_85_1_a4,
     author = {M. O. Korpusov and A. A. Panin and A. E. Shishkov},
     title = {On the critical exponent ``instantaneous~blow-up'' versus ``local solubility'' in the {Cauchy~problem} for a~model equation of {Sobolev} type},
     journal = {Izvestiya. Mathematics },
     pages = {111--144},
     publisher = {mathdoc},
     volume = {85},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a4/}
}
TY  - JOUR
AU  - M. O. Korpusov
AU  - A. A. Panin
AU  - A. E. Shishkov
TI  - On the critical exponent ``instantaneous~blow-up'' versus ``local solubility'' in the Cauchy~problem for a~model equation of Sobolev type
JO  - Izvestiya. Mathematics 
PY  - 2021
SP  - 111
EP  - 144
VL  - 85
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a4/
LA  - en
ID  - IM2_2021_85_1_a4
ER  - 
%0 Journal Article
%A M. O. Korpusov
%A A. A. Panin
%A A. E. Shishkov
%T On the critical exponent ``instantaneous~blow-up'' versus ``local solubility'' in the Cauchy~problem for a~model equation of Sobolev type
%J Izvestiya. Mathematics 
%D 2021
%P 111-144
%V 85
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2021_85_1_a4/
%G en
%F IM2_2021_85_1_a4
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/