On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 79-131
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In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities $|\nabla u|^q$, $\partial_t|\nabla u|^q$, and $\partial^2_t|\nabla u|^q $ are considered. For the corresponding Cauchy problems, we obtain results on local-in-time unique solvability in the weak sense and on blow-up for a finite time. These three equations have the same critical exponent $q=3/2$ since weak solutions behave differently for $1$ and for $q>3/2$. By the method of nonlinear capacity proposed by S. I. Pokhozhaev, we obtain a priori estimates, which imply the absence of local and global weak solutions.
Keywords:
nonlinear Sobolev-type equation, blow-up, local solvability, nonlinear capacity, estimates of the blow-up time.
@article{INTO_2020_185_a8,
author = {M. O. Korpusov and G. I. Shlyapugin},
title = {On blow-up of solutions of the {Cauchy} problems for a class of nonlinear equations of ferrite theory},
journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
pages = {79--131},
publisher = {mathdoc},
volume = {185},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/INTO_2020_185_a8/}
}
TY - JOUR AU - M. O. Korpusov AU - G. I. Shlyapugin TI - On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory JO - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory PY - 2020 SP - 79 EP - 131 VL - 185 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/INTO_2020_185_a8/ LA - ru ID - INTO_2020_185_a8 ER -
%0 Journal Article %A M. O. Korpusov %A G. I. Shlyapugin %T On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory %J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory %D 2020 %P 79-131 %V 185 %I mathdoc %U http://geodesic.mathdoc.fr/item/INTO_2020_185_a8/ %G ru %F INTO_2020_185_a8
M. O. Korpusov; G. I. Shlyapugin. On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 79-131. http://geodesic.mathdoc.fr/item/INTO_2020_185_a8/