Abstract:
We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation
∂tb(u)−div{|σ(u)|m−2σ(u)}=f(x,t),δ(u):=∇u+k(b(u))→e,|→e|=1,\enskipm>1,
with a monotone nondecreasing continuous function b. Such equations arise in the theory of non-Newtonian filtration as well as in the mathematical glaciology.
Citation:
A. V. Ivanov, J.-F. Rodrigues, “Existence and uniqueness of a weak solution to the initial mixt boundary value problem for quasilinear
elliptic-parabolic equations”, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Zap. Nauchn. Sem. POMI, 259, POMI, St. Petersburg, 1999, 67–88; J. Math. Sci. (New York), 109:5 (2002), 1851–1866
\Bibitem{IvaRod99}
\by A.~V.~Ivanov, J.-F.~Rodrigues
\paper Existence and uniqueness of a weak solution to the initial mixt boundary value problem for quasilinear
elliptic-parabolic equations
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~30
\serial Zap. Nauchn. Sem. POMI
\yr 1999
\vol 259
\pages 67--88
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1051}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1754358}
\zmath{https://zbmath.org/?q=an:0983.35094}
\transl
\jour J. Math. Sci. (New York)
\yr 2002
\vol 109
\issue 5
\pages 1851--1866
\crossref{https://doi.org/10.1023/A:1014488123746}
Linking options:
https://www.mathnet.ru/eng/znsl1051
https://www.mathnet.ru/eng/znsl/v259/p67
This publication is cited in the following 7 articles:
Antontsev S., Chipot M., Shmarev S., “Uniqueness and Comparison Theorems for Solutions of Doubly Nonlinear Parabolic Equations with Nonstandard Growth Conditions”, Commun. Pure Appl. Anal, 12:4 (2013), 1527–1546
Kiyeon Shin, Sujin Kang, “DOUBLY NONLINEAR VOLTERRA EQUATIONS INVOLVING THE LERAY-LIONS OPERATORS”, East Asian mathematical journal, 29:1 (2013), 69
Michal Beneš, “A Note on Doubly Nonlinear Parabolic Systems with Unilateral Constraint”, Results. Math., 63:1-2 (2013), 47
Antontsev S., Shmarev S., “Elliptic equations with triple variable nonlinearity”, Complex Variables and Elliptic Equations, 56:7–9 (2011), 573–597
Antontsev S., Shmarev S., “Parabolic equations with double variable nonlinearities”, Math Comput Simulation, 81:10 (2011), 2018–2032
Kang S., “Doubly nonlinear Volterra equations related to the p-Laplacian operator”, Math Methods Appl Sci, 34:9 (2011), 1065–1074
Antontsev S.N., De Oliveira H.B., “On a mathematical model in ice sheet dynamics”, Proceedings of the 5th Iasme/Wseas International Conference on Fluid Mechanics and Aerodynamics (Fma '07), Mathematics and Computers in Science and Engineering, 2007, 1–8