Existence of solution for some quasilinear parabolic systems with weight and weak monotonicity
DOI:
https://doi.org/10.22199/issn.0717-6279-2020-03-0033Keywords:
Nonlinear paraboliic system, Young measure, The divcurl type inequalityAbstract
We prove the existence of weak solution u for the nonlinear parabolic systems:
which is a Dirichlet Problem. In this system, v belongs to , f and g satisfy some standards continuity and growth conditions. We prove existence of a weak solution of different variants of this system under classical regularity for some growth and coercivity for σ but with only very mild monotonicity assumptions.
References
Y. Akdim, E. Azroul, and A. Benkirane, “Existence of solution for quasilinear degenerated elliptic equations”, Electronic journal of differential equations, vol. 2001, Art ID. 71, Nov. 2001. [On line]. Available: https://bit.ly/2ykRfMj
H. Brézis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: North-Holland, 1973.
F. Browder, “Existences theorems for nonlinear partial differential equations”, in Global analysis, vol. 16, S.-S.- Chern and S. Smale, Eds. Providence, RI: American Mathematical Society, 1970, pp. 8–67, doi: 10.1090/pspum/016
G. Dolzmann, N. Hungerbühler, and S. Müller, “Non-linear elliptic systems with measure-valued right hand side”, Mathematische zeitschrift, vol. 226, no. 4, pp. 545–574, Dec. 1997, doi: 10.1007/PL00004354
D. Blanchard and H. Redwane, “Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, natural growth terms and L1 data,” Arab Journal of Mathematical Sciences, vol. 20, no. 2, pp. 157–176, Jul. 2014, doi: 10.1016/j.ajmsc.2013.06.002
F. Augsburger, "Young measures and quasilinear systems in divergence form with weak monotonicity", PhD. thèse, Université de Fribourg, 2004.
G. B. Folland, Real analysis: modern techniques and applications. New York, NY: Wiley, 1973.
N. Hungerbühler, “Quasi-linear parabolic systems in divergence form with weak monotonicity”, Duke mathematical journal, vol. 107, no. 3, pp. 497–520, Mar. 2001, doi:10.1215/S0012-7094-01-10733-3
N. Hungerbüehler, Young measures and nonlinear PDEs. Birmingham, 1999. [On line] Available: https://bit.ly/3cL29d7
J. Kristensen, “Lower semicontinuity in spaces of weakly differentiable functions”, Mathematische annalen, vol. 313, no. 4, pp. 653–710, Apr. 1999. doi: 10.1007/s002080050277
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod, 1969.
J. Simon, “Compact sets in the spaceLp (O,T; B)”, Annali di matematica pura ed applicata, vol. 146, no. 1, pp. 65–96, Dec. 1986, doi: 10.1007/BF01762360
M. Valadier, “Young measures,” in Methods of nonconvex analysis, vol. 1146, A. Cellina, Ed. Berlin: Springer, 1990, pp. 152–188, doi: 10.1007/BFb0084935
M. Valadier, “A course on Young measures”, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 26 supp, pp. 349-394, 1994. [On line]. Available: https://bit.ly/3g8VNX7
K. Yosida, Functional analysis, 6th ed. Berlin: Springer, 1980.
E. Zeidler, Nonlinear functional analysis and its application, II/ A: linear monotone operators, New York, NY: Springer, 1990, doi: 10.1007/978-1-4612-0985-0
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Copyright (c) 2020 Azroul Elhoussine, Barbara Abdelkrim, Rami El Houcine
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