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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2020, Volume 185, Pages 79–131 (Mi into704)

On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory

M. O. Korpusov, G. I. Shlyapugin

Lomonosov Moscow State University

Abstract: 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 $1<q\leq 3/2$ 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

DOI: https://doi.org/10.36535/0233-6723-2020-185-79-131

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UDC: 517.538
MSC: 35B44

Citation: M. O. Korpusov, G. I. Shlyapugin, “On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory”, 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, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 185, VINITI, Moscow, 2020, 79–131

Citation in format AMSBIB
\Bibitem{KorShl20} \by M.~O.~Korpusov, G.~I.~Shlyapugin \paper On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory \inbook 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 \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2020 \vol 185 \pages 79--131 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into704} \crossref{https://doi.org/10.36535/0233-6723-2020-185-79-131}