Geodesic
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. 
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     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/}
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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/