Abstract:
This paper considers a boundary value problem for a quasilinear parabolic equation. In terms of Sobolev and Besov spaces the author determines a solution space A and a space B of initial conditions and right hand members such that the operator corresponding to the boundary value problem is a diffeomorphism, analytic in the Frechet sense, of the whole space A and a domain O in the space B. The behavior of the inverse operator of the problem around the boundary of O is studied, and it is shown that for different problems the domain O can coincide with the whole function space or be a strict subset of it.
Bibliography: 13 titles.
\Bibitem{Kuk82}
\by S.~B.~Kuksin
\paper Diffeomorphisms of function spaces corresponding to quasilinear parabolic equations
\jour Math. USSR-Sb.
\yr 1983
\vol 45
\issue 3
\pages 359--378
\mathnet{http://mi.mathnet.ru/eng/sm2213}
\crossref{https://doi.org/10.1070/SM1983v045n03ABEH001012}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=648413}
\zmath{https://zbmath.org/?q=an:0549.35067|0501.35046}
Linking options:
https://www.mathnet.ru/eng/sm2213
https://doi.org/10.1070/SM1983v045n03ABEH001012
https://www.mathnet.ru/eng/sm/v159/i3/p359
This publication is cited in the following 8 articles:
Sergei Kuksin, Vahagn Nersesyan, Armen Shirikyan, “Exponential mixing for a class of dissipative PDEs with bounded degenerate noise”, Geom. Funct. Anal., 30:1 (2020), 126
D. A. Khrychev, “Moments of solutions of evolution equations and suboptimal programmed
controls”, Sb. Math., 198:7 (2007), 1025–1062
Shirikyan A., “Qualitative Properties of Stationary Measures for Three-Dimensional Navier–Stokes Equations”, J. Funct. Anal., 249:2 (2007), 284–306
Agrachev A., Kuksin S., Sarychev A., Shirikyan A., “On Finite-Dimensional Projections of Distributions for Solutions of Randomly Forced 2D Navier–Stokes Equations”, Ann. Inst. Henri Poincare-Probab. Stat., 43:4 (2007), 399–415
Studies in Mathematics and Its Applications, 25, Attractors of Evolution Equations, 1992, 505
Yu. A. Dubinskii, “Higher-order nonlinear parabolic equations”, J. Soviet Math., 56:4 (1991), 2557–2607
D. A. Khrychev, “On approximate solution of systems of moment equations”, Math. USSR-Sb., 54:1 (1986), 81–98
Galaktionov V., Kurdyumov S., Samarskii A., “A Parabolic-System of Quasilinear Equations .1.”, Differ. Equ., 19:12 (1983), 1558–1574
Citation in format AMSBIB
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