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 Tr. Mat. Inst. Steklova, 2002, Volume 236, Pages 354–370 (Mi tm307)

Localized Boundary Blow-up Regimes for General Quasilinear Divergent Parabolic Equations of Arbitrary Order

A. E. Shishkov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: A mixed nonhomogeneous Cauchy–Dirichlet problem is considered for a general quasilinear parabolic equation in the divergence form in the case when the boundary data have an unbounded blow-up at a finite moment $T$. The energy space of this equation is $L_{\infty ,\mathrm {loc}}(0,T;L_{q+1}(\Omega ))\cap L_{p+1,\mathrm {loc}}(0,T;W_{p+1}^m(\Omega ))$, $m\ge 1$, $p>q>0$. The asymptotic behavior of an arbitrary energy solution for $t\to T$ is studied. Sharp (in a sense) integral constraints are established for the blow-up rate of the boundary data which guarantee the localization of the singularity zone of a solution in a certain neighborhood of the boundary of a domain (S-regime) or on the boundary itself (LS-regime).

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 341–356

Bibliographic databases:
UDC: 517.956.4

Citation: A. E. Shishkov, “Localized Boundary Blow-up Regimes for General Quasilinear Divergent Parabolic Equations of Arbitrary Order”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 354–370; Proc. Steklov Inst. Math., 236 (2002), 341–356

Citation in format AMSBIB
\Bibitem{Shi02} \by A.~E.~Shishkov \paper Localized Boundary Blow-up Regimes for General Quasilinear Divergent Parabolic Equations of Arbitrary Order \inbook Differential equations and dynamical systems \bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko \serial Tr. Mat. Inst. Steklova \yr 2002 \vol 236 \pages 354--370 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm307} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931037} \zmath{https://zbmath.org/?q=an:1125.35369} \transl \jour Proc. Steklov Inst. Math. \yr 2002 \vol 236 \pages 341--356