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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

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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

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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

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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

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

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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

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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

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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

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