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DOUBLY NONLINEAR PARABOLIC EQUATIONS RELATED TO THE LERAY-LIONS OPERATORS: TIME-DISCRETIZATION

  • Shin, Ki-Yeon (DEPARTMENT OF MATHEMATICS PUSAN NATIONAL UNIVERSITY) ;
  • Kang, Su-Jin (DEPARTMENT OF MATHEMATICS PUSAN NATIONAL UNIVERSITY)
  • Received : 2009.12.08
  • Accepted : 2010.03.10
  • Published : 2010.05.31

Abstract

In this paper, we consider a doubly nonlinear parabolic equation related to the Leray-Lions operator with Dirichlet boundary condition and initial data given. By exploiting a suitable implicit time-discretization technique, we obtain the existence of global strong solution.

Keywords

References

  1. R. Adams, Sovolev Spaces, Academic Press, New York, 1975.
  2. V. Barbu, Nonlinear semigroups and dierential equations in Banach spaces, Noordho Internat. Publ., Leyden, 1976.
  3. A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear P.D.E. having natural growth terms and unbounded solutions, Ann. Inst. H. Poincare. 5 (1988), no. 4, 347-364.
  4. A. Eden, B. Michaux and J. M. Rakotoson, Semidiscretized nonlinear evolution equa- tions as discrete dynamical systems and error analysis, Indiana University Mathematics J. 39 (1990), no. 3, 737-783 https://doi.org/10.1512/iumj.1990.39.39036
  5. A. Eden, B. Michaux and J. M. Rakotoson, Doubly nonlinear parabolic type equations as dynamical systems, J. Dynam. Di. Equ. 3 (1991), no. 1, 87-131. https://doi.org/10.1007/BF01049490
  6. A. El Hachimi and H. El. Ouardi, Existence and regularity of a global attractor for doubly nonlinear parabolic equations, Electronic J. Di. Equ. 2002 (2002), no. 45, 1-15.
  7. M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a mi- croforce balance, Phys.D 92 (1996), 178-192. https://doi.org/10.1016/0167-2789(95)00173-5
  8. V. Le and K. Schmit, Global Bifurcation in Variational Inequalities, Springer-Verlag, New York. Inc. 1997.
  9. A. Miranville and G. Schimperna, Global solution to a phase transition model based on microforce balance, J. evol. equ. 5 (2005), 253-276. https://doi.org/10.1007/s00028-005-0187-x
  10. A. Rougirel, Convergence to steady state and attractors for doubly nonlinear equations, J. Math. Anal. Appl. 339 (2008), 281-294. https://doi.org/10.1016/j.jmaa.2007.06.028
  11. J. M. Rakotoson, On some degenerate and nondegenerate Quasilinear Elliptic systems with nonhomogeneous Dirichlet boundary condition, Nonl. Anal. T.M.A. 13 (1989), no. 2, 165-183. https://doi.org/10.1016/0362-546X(89)90042-4
  12. M. Schatzman, Stationary solutions and asymptotic behavior of a Quasilinear degener- ate parabolic equation, Indiana Univ. Math. J. 33 (1984), no. 1, 1-29. https://doi.org/10.1512/iumj.1984.33.33001
  13. R. E. Showalter, Monotone operators in Banach space and nonlinear partial dierential equations, Math. Surveys and Monographs 49, Amer. Math. Soc., 1996.

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  1. DOUBLY NONLINEAR VOLTERRA EQUATIONS INVOLVING THE LERAY-LIONS OPERATORS vol.29, pp.1, 2013, https://doi.org/10.7858/eamj.2013.006