On Long-Time Dynamics of the Solution of Doubly Nonlinear Equation

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• 作者：Emil Novruzov ; Ali Hagverdiyev
• 关键词：Nonlinear degenerate equation ; Global attractor ; Finite speed of propagation of perturbations (FSP)
• 刊名：Qualitative Theory of Dynamical Systems
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：15
• 期：1
• 页码：127-155
• 全文大小：595 KB
• 参考文献：1.Ahmed, N., Sunada, D.K.: Nonlinear flows in porous media. J. Hydraul. Div. Proc. Am. Soc. Civil Eng. 95, 1847–1857 (1969)
  2.Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)MathSciNet CrossRef MATH
  3.Arai, T.: On the existence of the solution for \(\partial \varphi \left( u^{\prime }\left( t\right) \right) +\partial \psi \left( u\left( t\right) \right) \ni f\left( t\right) \) . J. Fac.Sci. Univ. Tokyo Sect. IA Math. 26, 75–96 (1979)MathSciNet
  4.Bamberger, A.: Etude d’une ‘ equation doublement non lin ’ eaire. J. Funct. Anal. 24, 148–155 (1977)MathSciNet CrossRef MATH
  5.Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden (1976)CrossRef MATH
  6.Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)MATH
  7.Bertsch, M., Kersner, R., Peletier, L.A.: Positivity versus localization in degenerate diffusion equations. Nonlinear Anal. 9, 987–1008 (1985)MathSciNet CrossRef MATH
  8.Blanchard, D., Francfort, G.A.: Study of double nonlinear heat equation with no growth assumptions on the parabolic term. SIAM J. Math. Anal. 9(5), 1032–1056 (1988)
  9.Brezis, H.: Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)MATH
  10.Colli, P.: On some doubly nonlinear evolution equations in Banach spaces. Jpn J. Ind. Appl. Math. 9, 181–203 (1992)MathSciNet CrossRef MATH
  11.Colli, P., Visintin, A.: On a class of doubly nonlinear evolution problems. Commun. Part. Differ. Equ. 15, 737–756 (1990)MathSciNet CrossRef MATH
  12.DiBenedetto, E., Showalter, R.E.: Implicit degenerate evolution equations and applications. SIAM J. Math. Anal. 12, 731–751 (1981)MathSciNet CrossRef MATH
  13.Eden, A., Michaux, B., Rakotoson, J.-M.: Doubly nonlinear parabolic-type equations as dynamical systems. J. Dyn. Differ. Equ. 3, 87–131 (1991)MathSciNet CrossRef MATH
  14.Eden, A., Rakotoson, J.-M.: Exponential attractors for a doubly non-linear equation. J. Math. Anal. Appl. 185, 321–339 (1994)MathSciNet CrossRef MATH
  15.Efendiev, M., Zelik, S.: Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations. Math. Methods Appl. Sci. 32, 1638–1668 (2009)MathSciNet CrossRef MATH
  16.Esteban, J.R., Vazquez, J.L.: Homogeneous diffusion in \(R\) with power-like nonlinear diffusivity. Arch. Ration. Mech. Anal. 103, 39–88 (1988)MathSciNet CrossRef MATH
  17.Ferreira, R., de Pablo, A., Reyes, G., Sánchez, A.: The interfaces of an inhomogeneous porous medium equation with convection. Commun. Part. Differ. Equ. 31, 497–514 (2006)MathSciNet CrossRef MATH
  18.Gentile, C.B., Primo, M.: Parameter dependent quasi-linear parabolic equations. Nonlinear Anal. 59, 801–812 (2004)MathSciNet CrossRef MATH
  19.Gilding, B.H.: The soil-moisture zone in a physically-based hydrologic model. Adv. Water Resour. 6, 36–43 (1983)CrossRef
  20.Grange, O., Mignot, F.: Sur la résolution d’une équation et d’une inéquation paraboliques non linéaires. J. Funct. Anal. 11, 77–92 (1972)MathSciNet CrossRef MATH
  21.Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83(1), 22–37 (1997)MathSciNet CrossRef
  22.Kamin, S., Rosenau, P.: Propagation of thermal waves in an inhomogenous medium. Commun. Pure Appl. Math. 34, 831–852 (1981)MathSciNet CrossRef MATH
  23.Kersner, R., Reyes, G., Tesei, A.: On class of parabolic equations with variable density and absorption. Adv. Differ. Equ. 7, 155–176 (2002)MathSciNet MATH
  24.Khanmamedov, AKh: Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain. J. Math. Anal. Appl. 316(2), 601–615 (2006)MathSciNet CrossRef MATH
  25.Lions, J.-L.: Quelques Methodes de Resolution des Problemes Aux Limites Nonlineaires. Dunod, Gauthier Villars, Paris (1969)MATH
  26.Miranville, A.: Finite dimensional global attractor for a class of doubly nonlinear parabolic equations. Cent. Eur. J. Math. 4(1), 163–182 (2006)MathSciNet CrossRef MATH
  27.Novruzov, E.: On long-time behavior of the positive solutions of nonhomogeneous non-Newtonian equation. Appl. Anal. 90(7), 1159–1168 (2011)MathSciNet CrossRef MATH
  28.Raviart, P.-A.: Sur la résolution de certaines équations paraboliques non linéaires. J. Funct. Anal. 5, 299–328 (1970)MathSciNet CrossRef MATH
  29.Segatti, A.: Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete Contin. Dyn. Syst. 14(4), 801–820 (2006)MathSciNet CrossRef MATH
  30.Shirakawa, K.: Large time behavior for doubly nonlinear systems generated by subdifferentials. Adv. Math. Sci. Appl. 10, 77–92 (2000)MathSciNet MATH
  31.Showalter, R.E.: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, vol 49. American Mathematical Society, Providence, RI (1997)
  32.Takeuchi, S., Yokota, T.: Global attractors for a class of degenerate diffusion equations. Electron. J. Differ. Equ. 76, 1–13 (2003)
  33.Tedeev, A.F.: Conditions for the time clobal existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations. Sib. Math. J. 45, 155–164 (2004)MathSciNet CrossRef
  34.Tedeev, A.F.: The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations. Appl. Anal. 86(6), 755–782 (2007)MathSciNet CrossRef MATH
  35.Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci., vol. 68, Springer, New-York (1988)
  36.Tsutsumi, M.: On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J. Math. Anal. Appl. 132, 187–212 (1988)MathSciNet CrossRef MATH
  37.Xiang, Z., Mu, Ch., Hu, X.: Support properties of solutions to a degenerate equation with absorption and variable density. Nonlinear Anal. 68(7), 1940–1953 (2008)MathSciNet CrossRef MATH
  38.Zhong, C.K., Yang, M.H., Sun, C.Y.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J. Differ. Equ. 223, 367–399 (2006)MathSciNet CrossRef MATH
  
• 作者单位：Emil Novruzov (1)
  Ali Hagverdiyev (2)
  
  1. Department of Mathematics, Hacettepe University, Ankara, Turkey
  2. Department of Applied Mathematics, Texas A&M University, College Station, TX, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Dynamical Systems and Ergodic Theory
  Difference and Functional Equations
  
• 出版者：Birkh盲user Basel
• ISSN：1662-3592

文摘

The subject of this investigation is the long time behavior of the positive solutions of the following nonhomogeneous equation $$\begin{aligned} \rho \left( \left| x\right| \right) \frac{\partial \beta \left( v\right) }{\partial t}-\overset{N}{\underset{i=1}{\sum }}D_{i}(|D_{i}v|^{ \lambda -1}D_{i}v)+\rho \left( \left| x\right| \right) g\left( \beta \left( v\right) \right) +l\beta ^{1+m}\left( v\right) =f\left( x\right) \qquad \end{aligned}$$ (1)in unbounded domain \( \mathbb {R} _{+}\times \mathbb {R} ^{N},\) where the term \(g\left( s\right) \) is supposed to satisfy a condition \(g^{\prime }\left( s\right) >-l_{1}\) and \(D_{i}=\partial _{x_{i}}\) . The existence of the global attractor for the Eq. (1) in \(L^{1+\theta }\left( \mathbb {R} ^{N},\rho \right) =\left\{ v;v\rho ^{1/(1+\theta )}\in L^{1+\theta }\left( \mathbb {R} ^{N}\right) \right\} \) is proved. Keywords Nonlinear degenerate equation Global attractor Finite speed of propagation of perturbations (FSP) Mathematics Subject Classification 35K55 35B45 35B41 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (38) References1.Ahmed, N., Sunada, D.K.: Nonlinear flows in porous media. J. Hydraul. Div. Proc. Am. Soc. Civil Eng. 95, 1847–1857 (1969)2.Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)MathSciNetCrossRefMATH3.Arai, T.: On the existence of the solution for \(\partial \varphi \left( u^{\prime }\left( t\right) \right) +\partial \psi \left( u\left( t\right) \right) \ni f\left( t\right) \). J. Fac.Sci. Univ. Tokyo Sect. IA Math. 26, 75–96 (1979)MathSciNet4.Bamberger, A.: Etude d’une ‘ equation doublement non lin ’ eaire. J. Funct. Anal. 24, 148–155 (1977)MathSciNetCrossRefMATH5.Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden (1976)CrossRefMATH6.Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)MATH7.Bertsch, M., Kersner, R., Peletier, L.A.: Positivity versus localization in degenerate diffusion equations. Nonlinear Anal. 9, 987–1008 (1985)MathSciNetCrossRefMATH8.Blanchard, D., Francfort, G.A.: Study of double nonlinear heat equation with no growth assumptions on the parabolic term. SIAM J. Math. Anal. 9(5), 1032–1056 (1988)9.Brezis, H.: Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)MATH10.Colli, P.: On some doubly nonlinear evolution equations in Banach spaces. Jpn J. Ind. Appl. Math. 9, 181–203 (1992)MathSciNetCrossRefMATH11.Colli, P., Visintin, A.: On a class of doubly nonlinear evolution problems. Commun. Part. Differ. Equ. 15, 737–756 (1990)MathSciNetCrossRefMATH12.DiBenedetto, E., Showalter, R.E.: Implicit degenerate evolution equations and applications. SIAM J. Math. Anal. 12, 731–751 (1981)MathSciNetCrossRefMATH13.Eden, A., Michaux, B., Rakotoson, J.-M.: Doubly nonlinear parabolic-type equations as dynamical systems. J. Dyn. Differ. Equ. 3, 87–131 (1991)MathSciNetCrossRefMATH14.Eden, A., Rakotoson, J.-M.: Exponential attractors for a doubly non-linear equation. J. Math. Anal. Appl. 185, 321–339 (1994)MathSciNetCrossRefMATH15.Efendiev, M., Zelik, S.: Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations. Math. Methods Appl. Sci. 32, 1638–1668 (2009)MathSciNetCrossRefMATH16.Esteban, J.R., Vazquez, J.L.: Homogeneous diffusion in \(R\) with power-like nonlinear diffusivity. Arch. Ration. Mech. Anal. 103, 39–88 (1988)MathSciNetCrossRefMATH17.Ferreira, R., de Pablo, A., Reyes, G., Sánchez, A.: The interfaces of an inhomogeneous porous medium equation with convection. Commun. Part. Differ. Equ. 31, 497–514 (2006)MathSciNetCrossRefMATH18.Gentile, C.B., Primo, M.: Parameter dependent quasi-linear parabolic equations. Nonlinear Anal. 59, 801–812 (2004)MathSciNetCrossRefMATH19.Gilding, B.H.: The soil-moisture zone in a physically-based hydrologic model. Adv. Water Resour. 6, 36–43 (1983)CrossRef20.Grange, O., Mignot, F.: Sur la résolution d’une équation et d’une inéquation paraboliques non linéaires. J. Funct. Anal. 11, 77–92 (1972)MathSciNetCrossRefMATH21.Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83(1), 22–37 (1997)MathSciNetCrossRef22.Kamin, S., Rosenau, P.: Propagation of thermal waves in an inhomogenous medium. Commun. Pure Appl. Math. 34, 831–852 (1981)MathSciNetCrossRefMATH23.Kersner, R., Reyes, G., Tesei, A.: On class of parabolic equations with variable density and absorption. Adv. Differ. Equ. 7, 155–176 (2002)MathSciNetMATH24.Khanmamedov, AKh: Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain. J. Math. Anal. Appl. 316(2), 601–615 (2006)MathSciNetCrossRefMATH25.Lions, J.-L.: Quelques Methodes de Resolution des Problemes Aux Limites Nonlineaires. Dunod, Gauthier Villars, Paris (1969)MATH26.Miranville, A.: Finite dimensional global attractor for a class of doubly nonlinear parabolic equations. Cent. Eur. J. Math. 4(1), 163–182 (2006)MathSciNetCrossRefMATH27.Novruzov, E.: On long-time behavior of the positive solutions of nonhomogeneous non-Newtonian equation. Appl. Anal. 90(7), 1159–1168 (2011)MathSciNetCrossRefMATH28.Raviart, P.-A.: Sur la résolution de certaines équations paraboliques non linéaires. J. Funct. Anal. 5, 299–328 (1970)MathSciNetCrossRefMATH29.Segatti, A.: Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete Contin. Dyn. Syst. 14(4), 801–820 (2006)MathSciNetCrossRefMATH30.Shirakawa, K.: Large time behavior for doubly nonlinear systems generated by subdifferentials. Adv. Math. Sci. Appl. 10, 77–92 (2000)MathSciNetMATH31.Showalter, R.E.: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, vol 49. American Mathematical Society, Providence, RI (1997)32.Takeuchi, S., Yokota, T.: Global attractors for a class of degenerate diffusion equations. Electron. J. Differ. Equ. 76, 1–13 (2003)33.Tedeev, A.F.: Conditions for the time clobal existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations. Sib. Math. J. 45, 155–164 (2004)MathSciNetCrossRef34.Tedeev, A.F.: The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations. Appl. Anal. 86(6), 755–782 (2007)MathSciNetCrossRefMATH35.Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Appl. Math. Sci., vol. 68, Springer, New-York (1988)36.Tsutsumi, M.: On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J. Math. Anal. Appl. 132, 187–212 (1988)MathSciNetCrossRefMATH37.Xiang, Z., Mu, Ch., Hu, X.: Support properties of solutions to a degenerate equation with absorption and variable density. Nonlinear Anal. 68(7), 1940–1953 (2008)MathSciNetCrossRefMATH38.Zhong, C.K., Yang, M.H., Sun, C.Y.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J. Differ. Equ. 223, 367–399 (2006)MathSciNetCrossRefMATH About this Article Title On Long-Time Dynamics of the Solution of Doubly Nonlinear Equation Journal Qualitative Theory of Dynamical Systems Volume 15, Issue 1 , pp 127-155 Cover Date2016-04 DOI 10.1007/s12346-015-0153-0 Print ISSN 1575-5460 Online ISSN 1662-3592 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Dynamical Systems and Ergodic Theory Difference and Functional Equations Keywords Nonlinear degenerate equation Global attractor Finite speed of propagation of perturbations (FSP) 35K55 35B45 35B41 Authors Emil Novruzov (1) Ali Hagverdiyev (2) Author Affiliations 1. Department of Mathematics, Hacettepe University, Ankara, Turkey 2. Department of Applied Mathematics, Texas A&M University, College Station, TX, USA Continue reading... To view the rest of this content please follow the download PDF link above.