American Institute of Mathematical Sciences

Advanced
July  2013, 12(4): 1755-1768. doi: 10.3934/cpaa.2013.12.1755

On the temporal decay estimates for the degenerate parabolic system

Tariel Sanikidze 1, and  A.F. Tedeev 2,

1. 

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, str. R. Luxemburg 74, Donetsk, 83114, Ukraine
2. 

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roza Luxemburg st.74, 340114 Donetsk

Received  March 2011 Revised  March 2012 Published  November 2012

We study long-time behavior for the Cauchy problem of degenerate parabolic system which in the scalar case coincides with classical porous media equation. Sharp bounds of the decay in time estimates of a solution and its size of support were established. Moreover, local space-time estimates under the optimal assumption on initial data were proven.
Keywords: Cauchy problem, parabolic systems, sharp estimates, local behavior..
Mathematics Subject Classification: Primary: 35K45; Secondary: 35B40, 35K6.
Citation: Tariel Sanikidze, A.F. Tedeev. On the temporal decay estimates for the degenerate parabolic system. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1755-1768. doi: 10.3934/cpaa.2013.12.1755
References:
[1]

D. Andreucci and E. Di Benedetto, A new approach to initial traces in nonlinear filtration, Annales Institut H. Poincaré Analyse non Linéaire, 7 (1990), 305-334. Google Scholar

[2]

D. Andreucci and A. F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Advances in Differential Equations, 5 (2000), 833-860. Google Scholar

[3]

D. Andreucci and A. F. Tedeev, Finite speed of propagation for thin film equations and other higher order parabolic equations with general nonlinearity, Interfaces and Free Boundaries, 3 (2001), 233-264. Google Scholar

[4]

D. Andreucci and A. F. Tedeev, Universal bounds at the blow-up time for nonlinear parabolic equations, Advances in Differential Equations, 10 (2005), 89-120. Google Scholar

[5]

S. Antontsev, J. I. Díaz and S. Shmarev, Energy Methods for Free Boundary Problems: Applications to Non-linear, in "PDEs and Fluid Mechanics," Bikhäuser, Boston, 2002, Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.  Google Scholar

[6]

Ph. Benilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $R^N$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87. Google Scholar

[7]

E. Di Benedetto, "Degenerate Parabolic Equations," Springer-Verlag, 1993. Google Scholar

[8]

E. Di Benedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, Journal für die reine und angewandte Math., 357 (1985), 1-22. Google Scholar

[9]

A. Jüngel, P. A. Markovich and G. Toscani, Decay rate for solutions of degenerate parabolic systems, J. Diff. Eqns., Conf. 06, (2001), 189-202. Google Scholar

[10]

A. S. Kalashnikov, Some properties of the qualitative theory of nonlinear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 169-222. Google Scholar

[11]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," volume 23 of Translation of Mathematical Monographs, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[12]

J. L. Vazquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs. Clarendon Press, Oxvord, 2007.  Google Scholar

[13]

H. M. Yin, On p-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory, Quart. appl. Math., 59 (2001), 47-66. Google Scholar

[14]

H. M. Yin, A degenerate evolution system modelling Bean's critical-state type-II superconductors, Discrete and Continuous Dynamical Systems, 8 (2002), 781-794. Google Scholar

[15]

H. M. Yin, On degenerate parabolic system, J. Differential Equations, 245 (2008), 722-736. Google Scholar

[16]

H. Yuan, The Cauchy problem for a quasilinear degenerate parabolic system, Nonlinear Analysis, TMA, 23 (1994), 155-164. Google Scholar

show all references

References:
[1]

D. Andreucci and E. Di Benedetto, A new approach to initial traces in nonlinear filtration, Annales Institut H. Poincaré Analyse non Linéaire, 7 (1990), 305-334. Google Scholar

[2]

D. Andreucci and A. F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Advances in Differential Equations, 5 (2000), 833-860. Google Scholar

[3]

D. Andreucci and A. F. Tedeev, Finite speed of propagation for thin film equations and other higher order parabolic equations with general nonlinearity, Interfaces and Free Boundaries, 3 (2001), 233-264. Google Scholar

[4]

D. Andreucci and A. F. Tedeev, Universal bounds at the blow-up time for nonlinear parabolic equations, Advances in Differential Equations, 10 (2005), 89-120. Google Scholar

[5]

S. Antontsev, J. I. Díaz and S. Shmarev, Energy Methods for Free Boundary Problems: Applications to Non-linear, in "PDEs and Fluid Mechanics," Bikhäuser, Boston, 2002, Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.  Google Scholar

[6]

Ph. Benilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $R^N$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87. Google Scholar

[7]

E. Di Benedetto, "Degenerate Parabolic Equations," Springer-Verlag, 1993. Google Scholar

[8]

E. Di Benedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, Journal für die reine und angewandte Math., 357 (1985), 1-22. Google Scholar

[9]

A. Jüngel, P. A. Markovich and G. Toscani, Decay rate for solutions of degenerate parabolic systems, J. Diff. Eqns., Conf. 06, (2001), 189-202. Google Scholar

[10]

A. S. Kalashnikov, Some properties of the qualitative theory of nonlinear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 169-222. Google Scholar

[11]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," volume 23 of Translation of Mathematical Monographs, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[12]

J. L. Vazquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs. Clarendon Press, Oxvord, 2007.  Google Scholar

[13]

H. M. Yin, On p-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory, Quart. appl. Math., 59 (2001), 47-66. Google Scholar

[14]

H. M. Yin, A degenerate evolution system modelling Bean's critical-state type-II superconductors, Discrete and Continuous Dynamical Systems, 8 (2002), 781-794. Google Scholar

[15]

H. M. Yin, On degenerate parabolic system, J. Differential Equations, 245 (2008), 722-736. Google Scholar

[16]

H. Yuan, The Cauchy problem for a quasilinear degenerate parabolic system, Nonlinear Analysis, TMA, 23 (1994), 155-164. Google Scholar

[1]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541
[2]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[3]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021097
[4]

Todor Gramchev, Nicola Orrú. Cauchy problem for a class of nondiagonalizable hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 533-542. doi: 10.3934/proc.2011.2011.533
[5]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423
[6]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761
[7]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[8]

Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3
[9]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[10]

Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091
[11]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100
[12]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[13]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[14]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721
[15]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[16]

Horst Heck, Matthias Hieber, Kyriakos Stavrakidis. $L^\infty$-estimates for parabolic systems with VMO-coefficients. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 299-309. doi: 10.3934/dcdss.2010.3.299
[17]

Tommaso Leonori, Martina Magliocca. Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131
[18]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783
[19]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541
[20]

David Cruz-Uribe, SFO, José María Martell, Carlos Pérez. Sharp weighted estimates for approximating dyadic operators. Electronic Research Announcements, 2010, 17: 12-19. doi: 10.3934/era.2010.17.12

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences