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Mathematics of the USSR-Sbornik, 1989, Volume 64, Issue 2, Pages 527–542
DOI: https://doi.org/10.1070/SM1989v064n02ABEH003326
(Mi sm1758)
 

This article is cited in 13 scientific papers (total in 13 papers)

Quasilinear parabolic equations containing a Volterra operator in the coefficients

G. I. Laptev
References:
Abstract: Conditions are established for solvability in the large of the first initial-boundary value problem in a bounded domain $\Omega\subset R^n$ for the equation
$$ u_t+(-1^m)\sum_{|\alpha|=m}D^\alpha\biggl[a_\alpha\biggl(\int_0^t|D^\alpha u|^q\,dt\biggr)|D^\alpha u|^{q-2}D^\alpha u\biggr]=f, $$
where $q\geqslant2$. It contains the integral of the unknown function in the coefficients. The problem is regarded as an evolution equation of the form $u'+Au=f$. Conditions of polynomial growth are imposed on the functions $a_\alpha(s)$:
$$ a_0s^r\leqslant a_\alpha(s)\leqslant a_1s^r+a_2\qquad(a_i>0;\ r>0). $$
The space $\mathring W_p^m(\Omega;L^q(0,T))$, is constructed, where $p=q(1+r)$; the operator $A$ is coercive in this space. Under the additional assumption that the functions $a_\alpha(s)$ are convex (which corresponds to exponents $0<r\leqslant1$) it is proved that $A$ is a monotone operator and the corresponding evolution equation is solvable.
Bibliography: 6 titles.
Received: 27.11.1987
Bibliographic databases:
UDC: 517.956.35
MSC: Primary 35K22, 35K55; Secondary 45D05, 47H06, 47H07
Language: English
Original paper language: Russian
Citation: G. I. Laptev, “Quasilinear parabolic equations containing a Volterra operator in the coefficients”, Math. USSR-Sb., 64:2 (1989), 527–542
Citation in format AMSBIB
\Bibitem{Lap88}
\by G.~I.~Laptev
\paper Quasilinear parabolic equations containing a Volterra operator in the coefficients
\jour Math. USSR-Sb.
\yr 1989
\vol 64
\issue 2
\pages 527--542
\mathnet{http://mi.mathnet.ru//eng/sm1758}
\crossref{https://doi.org/10.1070/SM1989v064n02ABEH003326}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=965891}
\zmath{https://zbmath.org/?q=an:0683.35042}
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  • https://doi.org/10.1070/SM1989v064n02ABEH003326
  • https://www.mathnet.ru/eng/sm/v178/i4/p530
  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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