# American Institute of Mathematical Sciences

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

## On the temporal decay estimates for the degenerate parabolic system

 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.
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
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##### 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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