Abstract
The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form \(-\,\mathrm{div}\, a(x,t,u,\nabla u)\). The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a \(\Delta _2\) condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.
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References
Adams, R.: Sobolev Spaces. Press, New York (1975)
Aharouch, L., Bennouna, J.: Existence and uniqueness of solutions of unilateral problems in Orlicz spaces. Nonlinear Anal. Theory Methods Appl. 72, 3553–3565 (2010)
Antontsev, S.N., Chipot, M.: The thermistor problem: existence, smoothness, uniqueness, blowup. SIAM J. Math. Anal. 25, 1128–1156 (1994)
Apushkinskaya, D., Bildhauer, M., Fuchs, M.: Steady states of anisotropic generalized Newtonian fluids. J. Math. Fluid Mech. 7, 261–297 (2005)
Azroul, E., Redwane, H., Rhoudaf, M.: Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces. Port. Math. 66(1), 29–63 (2009)
Bildhauer, M., Fuchs, M., Zhong, X.: On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids. Algebra i Analiz 18, 1–23 (2006) (Translation in St. Petersburg Math. J. 18 (2007), 183–199)
Dankert, G.: Sobolev Embedding Theorems in Orlicz Spaces. Ph.D. thesis, University of Köln (1966)
Diening, L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129(8), 657–700 (2005)
Donaldson, T.K., Trudinger, N.S.: Orlicz–Sobolev spaces and imbedding theorems. J. Funct. Anal. 8(1), 52–75 (1971)
Elmahi, A., Meskine, D.: Parabolic initial-boundary value problems in Orlicz spaces. Ann. Polon. Math. 85, 99–119 (2005)
Elmahi, A., Meskine, D.: Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces. Nonlinear Anal. Theory Methods Appl. 60, 1–35 (2005)
Elmahi, A., Meskine, D.: Strongly nonlinear parabolic equations with natural growth terms and \(L^1\) data in Orlicz spaces. Port. Math. N. S. 62(2), 143–183 (2005)
González Montesinos, M.T., Ortegón Gallego, F.: The evolution thermistor problem with degenerate thermal conductivity. Commun. Pure Appl. Anal. 1(3), 313–325 (2002)
González Montesinos, M.T., Ortegón Gallego, F.: Existence of a capacity solution to a coupled nonlinear parabolic-elliptic system. Commun. Pure Appl. Anal. 6(1), 23–42 (2007)
Gossez, J.P.: Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)
Gossez, J.P.: Some approximation properties in Orlicz–Sobolev. Stud. Math. 74, 17–24 (1982)
Hadj Nassar, S., Moussa, H., Rhoudaf, M.: Renormalized Solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz spaces. Appl. Math. Comput. 249, 253–264 (2014)
Krasnosel’skii, M.A., Rutickii, Y.B.: Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1969)
Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)
Rajagopal, K.R., Ru̇žička, M.: Mathematical modeling of electrorheological materials. Continue Mech. Thermodyn. 13, 59–78 (2001)
Ru̇žička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer, Berlin (2000)
Tienari, M.: A degree theory for a class of mappings of monotone type in Orlicz–Sobolev spaces. In: Annales Academie Scientiarum Fennice, Helsinki (1994)
Xu, X.: A degenerate Stefan-like problem with Joule’s heating. SIAM J. Math. Anal. 23, 1417–1438 (1992)
Xu, X.: A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type. Commun. Part. Differ. Equ. 18, 199–213 (1993)
Xu, X.: On the existence of bounded temperature in the thermistor problem with degeneracy. Nonlinear Anal. Theory Methods Appl. 42, 199–213 (2000)
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Moussa, H., Ortegón Gallego, F. & Rhoudaf, M. Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces. Nonlinear Differ. Equ. Appl. 25, 14 (2018). https://doi.org/10.1007/s00030-018-0505-y
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DOI: https://doi.org/10.1007/s00030-018-0505-y
Keywords
- Capacity solution
- Coupled system
- Nonlinear parabolic–elliptic equations
- Weak solutions
- Orlicz–Sobolev spaces
- Thermistor problem