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Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 3
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Articles

Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent

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Pages 524-544 | Received 24 Nov 2013, Accepted 09 Feb 2015, Published online: 16 Mar 2015

References

  • Zhikov VV. Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR-Izv. 1987;29:33–66.
  • Zhikov VV. On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 1995;3:249–269.
  • Rajagopal KR, Růz̆ic̆ka M. Mathematical modeling of electrorheological materials. Continu. Mech. Thermodyn. 2001;13:59–78.
  • Zhikov VV. Solvability of the three-dimensional thermistor problem. Proc. Stekolov Inst. Math. 2008;261:101–114.
  • Chen YM, Levine S, Rao M. Variable exponent linear growth functionals in image restoration. SIMA. J. Appl. Math. 2006;66:1383–1406.
  • Fu YQ, Pan N. Existence of solutions for nonlinear parabolic problems with p(x)-growth. J. Math. Anal. Appl. 2010;362:313–326.
  • Fu YQ, Pan N. Local boundedness of weak solutions for nonlinear parabolic problem with p(x)-growth. J. Ineq. Appl. 2010, Article ID 163296, 16pp.
  • Diening L, Harjulehto P, Hästö P, Růz̆ic̆ka M. Lebesgue and Sobolev spaces with variable exponents. Vol. 2017, Lecture notes in mathematics. Berlin: Springer; 2011.
  • Fu YQ, Xiang MQ, Pan N. Regularity of weak solutions for nonlinear parabolic problem with p(x)-growth. Eectron. J. Qual. Theo. Differ. Equ. 2012;4:1–26.
  • Diening L, Nägele P, Růz̆ic̆ka M. Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var. Elliptic Equ. 2012;57:1209–1231.
  • Buhrii OM, Lavrenyuk SP. On a parabolic variational inequality that generalizes the equation of polytropic filtration. Ukr. Math. J. 2001;53:1027–1042.
  • Buhrii OM, Mashiyev RA. Uniqueness of solutions of parabolic variational inequality with variable exponent of nonlinearity. Nonlinear Anal. 2009;70:2326–2331.
  • Antontsev S, Shmarev S. Parabolic equations with anisotropic nonstandard growth conditions. Free Bound. Probl. 2007;60:33–44.
  • Antontsev S, Shmarev S. Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 2009;53:355–399.
  • Kovácik O, Rákosník J. On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J. 1991;41:592–618.
  • Fan XL, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J. Math. Anal. Appl. 2001;263:424–446.
  • Gonnino M. Quasilinear degenerate parabolic equation of Kirchhoff type. Math. Methods Appl. Sci. 1999;22:375–388.
  • Chipot M, Lovat B. Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Dyn. Contin. Discrete Impuls. Syst. Ser. A. 2001;8:35–51.
  • Zheng S, Chipot M. Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 2005;45:301–312.
  • Tudorascu A, Wunsch M. On a nonlinear, nonlocal parabolic problem with conservation of mass, mean and variance. Comm. Partial Differ. Equ. 2011;36:1426–1454.
  • Ghist M, Gobbino M. Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates. J. Differ. Equ. 2008;245:2979–3007.
  • Hashimoto H, Yamazaki T. Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type. J. Differ. Equ. 2007;237:491–525.
  • Autuori G, Pucci P, Salvatori MC. Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 2009;352:149–165.
  • Autuori G, Pucci P. Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces. Complex Var. Elliptic Equ. 2011;56:715–753.
  • Lovat B. Etudes de quelques problèms paraboliques non locaux [On a class of non linear and non local parabolic problems] [Thèse]. Metz: Universite de Metz; 1995.
  • Limaco JL, Medeiros LA. Kirchhoff-carrier elastic strings in noncylindrical domains. Port. Math. 1999;14:464–500.
  • Ghergu M, Rădulescu V. Nonlinear PDEs: mathematical models in biology, chemistry and population genetics. Springer monographs in mathematics. Heidelberg: Springer Verlag; 2012.
  • Vázquez JL, Vitillaro E. Heat equation with dynamical boundary conditions of locally reactive type. Semigroup Forum. 2007;74:1–40.
  • Vázquez JL, Vitillaro E. Heat equation with dynamical boundary conditions of reactive type. Comm. Partial Differ. Equ. 2008;33:561–612.
  • Lions JL. Quelques Méthodes de Résolution des Problèmes aux Limites Nonlineaires [Some methods of nonlinear boundary value problems]. Paris: Dunod, Gauthier-Villars; 1969.
  • Landes R. On the existence of weak solutions for quasilinear parabolic initial boundary value problem. Proc. Roy. Soc. Edinburgh Sect. A. 1981;89:217–237.
  • Chabrowski J, Fu YQ. Existence of solutions for p(x)-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 2005;306:604–618. Erratum: J. Math. Anal. Appl. 2006;323:1483.
  • Adams RA. Sobolev spaces. New York (NY): Academic Press; 1975.

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